THE Relativity theory of Einstein whereby he seeks to explain the physical universe, may be considered in several aspects, most of which do not concern us in this place. We are not interested here either in the physics or metaphysics of relativity, nor in the epistemological problems that it gives rise to. We are not even interested in it as a mathematical problem in the general sense, but only in that phase of it which concerns geometry. In other words, we are for the moment solely concerned with its relation to what is known as Euclidean geometry.

When we go to the chief of the school and its real founder for information on this point, we find ourselves confronted with a difficulty. We find that both the mathematical and physical concepts of Einstein seem to be in such a state of flux and apparent uncertainty that we can scarcely draw anything precise from his writings as to the relation of Euclidean geometry to physical science. All that we can make out is that he himself does not employ its conclusions, and that his methods are otherwise.

He has expressed his ideas in his own writings on Relativity, notably in a lecture on Geometry and Experience delivered to the Prussian Academy of Sciences in Berlin, on January 27, 1921, as appearing in an English translation, Sidelights on Relativity, and also in the volume on the Special and General Theory of Relativity. It is no wonder that Einstein so often relegates the explanation of what should be his own ideas to others, for, if we are at all to judge by his own attempts at expressing them, they are by no means clear in his own mind, and he is himself struggling to give, in a hazy way, some kind of reality to his mathematics by clothing his formulae with some interpretation or other. Since he starts with certain assumptions, the interpretation must bear some connection with these assumptions, and there, we might almost say, clarity ends, and we step into a region of mistiness and fog. We certainly cannot consider Einstein as one who shines as a scientific discoverer in the domain of physics, but rather as one who in a fuddled sort of way is merely trying to find some meaning for mathematical formulae in which he himself does not believe too strongly, but which he is hoping against hope somehow to establish. No wonder he seeks help to elucidate what is none too lucid for himself.

The pronouncements on the subject in the essays mentioned are very unsatisfactory, even if we realize that a few thin thoughts had to be spread out over the length of a lecture. It would be equally unsatisfactory to follow the reasoning, or what passes for reasoning and sequence of thought in these lectures, but certain points stand forth in the whole matter. First, Einstein departs from Euclidean geometry, and adopts as a basis for his mathematical equations the Riemannian conception of space and space relations; secondly, he adopts the view of Minkowski of a four-dimensional continuum, and the Riemann analysis amended in this sense; thirdly, he adopts the modern view that geometry has nothing to do with reality, and is a purely logical-formal science. On this fact apparently he bases his whole system.

Let us first take his attitude on this last point, which is that from which he sets out. He takes over whole the view of the modern nominalists that geometry has directly no relation to reality. Here are his words: “How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things? In my opinion the answer to this question is, briefly, this;—As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality.”¹ Let us examine these questions and answers, in order to place before our minds the peculiar logic of Einstein. He really asks two questions, and mark them well. The first is, “How is it, that mathematics is so appropriate an instrument for the development of physical science.” We naturally expect as answer an explanation of how this appropriateness comes about, since the appropriateness seems to be conceded. Does Einstein answer in this way? By no means, as we shall see. The second question goes a step further, and asks if unaided reason without experience could create such an appropriate instrument. The answer expected should be either yes or no, with an explanation of his choice. Does he do this? No, again. He has forgotten both his questions, and answers something else. His answer is that as far as the laws of mathematics refer to reality, they are uncertain; as far as they are certain, they do not refer to reality. Is this not an interesting commentary on his mind? He asks one, or rather two questions, and answers a third entirely different. The question he answers would naturally have the form, “Is the connection of mathematics and reality certain.”

Now a man with a logical turn of mind, which is used to running on a straight road, could not be guilty of such crooks and turns. He would either have answered the questions he put, or put the question he answered. But has the answer no relation to the question? Doubtless it has, but only by implication. In the answer given he actually denies what he assumed as clear in the questions, but he does not want to say he is denying it, even to himself. In short, he really does not know what he can affirm or deny.

He starts out to lecture on Geometry and Experience by giving two reasons why mathematics enjoys special esteem. These are: (l) because its laws are absolutely certain and indisputable, while in the physical sciences they are debatable and liable to be overthrown by new discoveries; (2) because mathematics affords the physical sciences a certain measure of security which without it they could not attain. Here then he asks the famous two questions; each supposes an appropriate connection between mathematics and physics; he ends by answering something else which by implication denies what the questions and the introductory statement suppose.

Now Einstein (if logical) either accepts the two reasons or he does not accept them; he accepts what he allows from the way he puts his questions, or he does not. In either case he should say so. But in neither does he do so. In one place he writes absolutely as if he accepted them; in the other he refuses to accept them. He does not say these are the reasons alleged or urged by others, but gives them as his own. He puts questions not requiring affirmation or negation of these reasons, but of the how and why of them. Then in his answer he suddenly leaps to an entirely different frame of mind, and answers something else which, in final analysis, entirely negatives his former frame of mind; for, if mathematics are not certain as far as they refer to reality, they are of no use in the sciences, and cannot be appropriate instruments for their development.

We have entered into this analysis of his language merely to show the difficulty we have to encounter on every page of his writings, to know just what he holds and what he does not hold. For he really does not know himself. He only knows that he is a Relativist, or rather he calls himself such, but he is lost when he wants to define what it means, and tell what elements go to make up this condition of mind in relation to other knowledge. We might answer that Einstein has not a logical mind—which may be quite true; but that is not all; it is not a sufficient answer to explain his inconsistency. He is inconsistent chiefly because he holds contradictory, incongruous things; but he does not hold them clearly or vividly enough before his mind, to see their contradiction or incongruity. Everything is somehow made one and rendered dim in outline by a mental fog. For instance, he really does hold in some sort of a way, what he states first, that the mathematics are what he says they are in science, and he would not dare to come out straightforwardly and deny it. But he holds also that the mathematics are, as we shall see, a purely formal creation, and he answers his question in this frame of mind. But he has not the logic or the perspicacity to place the two opinions together, contrast them, and either iron out the apparent inconsistency or affirm it. Like the proverbial hen on a hot griddle he cannot stand on one foot long enough to come to any individual satisfactory conclusion with regard to his individual conclusions.

In the next place, we come to the strange philosophy and stranger epistemology contained in the answer, namely, that mathematics as far as they are certain, do not refer to reality. Apparently, he means that geometry is not built on experience; his questions, as well as what follows in the lecture, show that this is his mental attitude. But you can not construct the slightest idea or thing used in mathematics without experience; for instance, in geometry no human being could, without experience, understand what are distance and direction, the fundamental ideas of the science. Such a statement is then meaningless. Let us take it in a sense that might have meaning, that from experience the intellect constructs ideas that are so ideal that they cannot be applied to reality. This is a pure question of metaphysics, and not one for mathematics to deal with. Here we are in plain idealistic conceptualism; in other words, Einstein allows no realism to human knowledge, and therefore is a pure skeptic. His answer is that as far as we know, we are uncertain, and as far as our knowledge is certain, it does not apply to reality. He has gone farther than any conceptualist or nominalist; for these always try to show some connection with reality, but he denies the possibility of all connection. Knowledge would therefore be use less, his Relativity as well as all other kinds of knowledge.

What is this knowledge that does not apply to reality? Knowledge of what? It is either a knowledge of something or nothing. If it is a knowledge of nothing, it is no knowledge. If it is a knowledge of something, it is a knowledge of something real; the two are synonymous. Now real may be taken in two senses; as actually existing, and, therefore, having actual reality; again as real in the sense that there is no logical contradiction in it and it is really possible. Thus in one sense it is real, and in another it is not. If knowledge is real in the first sense, it certainly applies to reality; if it is real in the second sense, it is knowledge of reality the moment we can judge of the real existence of the possible. In what sense is geometrical knowledge real according to Einstein? His answer is that inasmuch as knowledge is real it is uncertain; inasmuch as it is certain it is unreal. We have the neat epistemological contradiction of the same bit of knowledge being certain and uncertain at the same time with regard to the real; for knowledge must have some kind of reality, or it is no knowledge. There is then only one possible interpretation to Einstein’s answer that would allow it any meaning at all; for, if the answer is not a mere huddle of meaningless words, it signifies that so far as knowledge applies to the possible real it is certain; so far as it applies to the existing real, it is uncertain. For instance, I can create the idea of a possible elephant, and then the knowledge I have of elephants is certain; but when I actually see an elephant, and apply my knowledge to the existing thing, it is uncertain. What Einstein would then deny is the certainty of experience. This would not tell us how we got certain knowledge of the possible elephant, since our experience of the actual elephant is uncertain. This is hardly what was in Einstein’s mind, but an analysis of his language will never show what is in his mind; we must guess it.

It is true that Einstein applies his principle only to mathematics. But if he refused it further extension he would merely be illogical. There is no more reason for denying certainty to our conception of extended quantity, than there is for denying certainty to discrete quantity. The same intellect is there acting in the same way, and the same quantity is there only under another form as the object of our sense experience. If, then, our ideas of extension do not apply to reality, why should our ideas that implicate number apply to reality; how, for instance, should our idea of man or animal be applicable to several men or several animals? In that case all science would become a crazy dream.

Perhaps Einstein did not mean this. Probably he did not. I should rather consider that he is treating subjects which he is not competent to treat, and that he does not know the force of his own language, or the necessary implications of his thought. This is precisely the difficulty in reading Einstein. He is so hazy in his thought, so obscure in his language, so involved and illogical in presenting it, that interpretation becomes largely a matter of guesswork. Let anyone who wishes to know what I mean, try to analyze strictly a page or so of Einstein; for instance, let him try to draw up his reasoning in strict syllogistic form.

In any case there is no doubt that he accepts the conclusion of his school that there is a difference between the logical-formal and the objective or intuitive geometry. By the former he means evidently a pure product of the intellect, by the latter our experience of real things. This matter cannot occupy us in a pure mathematical question, since it belongs to the philosopher; and here Einstein hardly shows that he is a metaphysical opponent worthy of the least consideration. Were one to treat the question, it is the leaders and expounders of the school that should come in for attention.

But let us see how he applies it to geometry.

“Geometry sets out from certain conceptions such as ‘plane,’ ‘point,’ ’straight line,’ with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which in virtue of these ideas, we are inclined to accept as ’true.’ . . . A proposition is then correct (“true”) when it has been derived in the recognized manner from the axioms. The question of the ’truth’ of the individual geometrical propositions is thus reduced to one of the “truth” of the axioms. Now it has long been known that the last question is not only unanswerable by the methods of geometry, but that it is in itself entirely without meaning. We cannot ask whether it is true that only one straight line goes through two points. We can only say that Euclidean geometry deals with things called ’straight lines,’ to each of which is ascribed the property of being uniquely determined by two points situated on it. The concept true does not tally with the conceptions of pure geometry, because by the word true we are eventually in the habit of designating always the correspondence with the “real” object; geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves.“²

On the other hand, geometry owes its very existence to the need which was felt of knowing the spatial relations of real things, and it is actually used to measure them. Einstein meets this by saying, “Geometrical ideas correspond to more or less exact objects in nature, and these last are undoubtedly the exclusive cause of the genesis of those ideas.” But this connection is looked upon as a fault which the science of geometry ought to avoid. “Geometry ought to refrain from such a course in order to give to its structure the largest possible logical unity.”³ We need not call attention to the exactness of the thought which here clearly says that objects in nature are the exclusive cause of the genesis of these ideas, while above he states⁴ that mathematics is a product of human thought independent of experience; and on the same page, that geometry treats of entities which do not take for granted any knowledge or intuition whatever, but presupposes only the validity of the axioms; and later: “These axioms are free creations of the human mind” ; and “In axiomatic geometry the words point, etc., stand only for empty conceptual schemata.”⁵ How all this can be reconciled with the fact that the exclusive cause of their genesis is real things, is one of the peculiar logical combinations of the Einsteinian mental whirligig; it depends entirely on the hobby horse he is riding for the moment.

Now, either experience is the cause of these notions, or it is not. They are either created by the pure intellect independent of experience, or they are not. Both cannot be true at the same time. The division is clear-cut and defined; the only difficulty is that Einstein’s mind is not clear enough to cut and separate them. He holds the one with the majority of the human race; he has received the other from his school of mathematical nominalists, and he tries to hold both together. Just as above, and the case is parallel, both are held so hazily that he can’t see just how they shape up the one opposite the other, nor can he see that they exclude one another.

Another peculiar combination of the same kind in the exposition of Einstein is that while these things are creations of the mind with no relation to reality, they correspond to more or less exact objects in nature.⁶ He does not seem to feel the need of explaining how there can be any correspondence at all, where there is no relation. He does not say whether the correspondence is formal or only accidental. It would have to be accidental, if his language above meant anything. Then the question would arise, what is the use of an accidental correspondence. If it is formal, one might as well freely create in his mind a huge mountain of gold, and then find a more or less exact object in nature to correspond to it. We do not think a few tons off one way or the other would worry most creators. This is just another case of Einstein’s vague mentality, so vague that one can never interpret his words strictly, or really say what he means; for what he means and actually says are very different things, even if he does not know it.

Such is the view put forward by Einstein and the other moderns, which is to replace classic mathematics and science, and they are not a bit diffident in telling us about its superiority. According to Einstein, “This view of axioms, advocated by modern ‘axiomatics’ purges mathematics of all extraneous elements, and thus dispels the mystic obscurity which formerly surrounded the principles of mathematics.”⁷ There certainly is humor in the passage, if nothing else. Just contrast the fogginess of Euclid with Einstein’s startling lucidity. Then the immense majority of men are simply morons.

We have seen above that Einstein allows some kind of correspondence between those wonderful ideas created ex nihilo sui et subjecti and real things, but this is not the reason for connecting them with reality. We have here another surprisingly non-mystic combination of ideas. Let us hear how he dispels all such mystic obscurity of correspondence.

“It is clear that the system of concepts of axiomatic geometry alone cannot make any assertions as to the relation of real objects of this kind, which we will call practically rigid bodies. To be able to make such assertions, geometry must be stripped of its merely logical-formal character by the coordination of real objects of experience with the empty conceptual framework of axiomatic geometry. To accomplish this we need only to add the proposition: Solid bodies are related with respect to their possible dispositions as are bodies in Euclidean geometry of three dimensions. Then the propositions of Euclid contain affirmations as to the relations of practically rigid bodies.”⁸

Without trying to visualize how the logical-formal character can be stripped from geometry by coordination of the real objects of experience with the empty conceptual framework of axiomatic geometry, or just what is the physical process, or metaphysical, of stripping by coordinating the thing to be stripped with something else, this much stands forth, that the coordination and stripping is performed by merely adding a proposition. Simple enough. No mystic obscurity. We seem to have heard something like that before in the song of creation:

Just a proposition and it is done. Surely simple and clear enough. The proposition is made, and behold the correspondence is there. Try it on a mountain of gold.

Let us hear another view.

“If we deny the relation between the body of axiomatic Euclidean geometry and the practically rigid body of reality, we readily arrive at the following view, which was entertained by that acute and profound thinker, H. Poincaré.”⁹

We are not interested in the “profound” thought, but in the expression introducing it; i.e., If we deny. In olden days people really thought they ought to discover something before they could lay claim to scientific achievement. The Copernicus, the Keplers, the Galileos, the Newtons, all added to the sum of scientific knowledge in one way or another by learning something new from reality. Their modern rivals have found a much simpler and less mystic way. All they have to do is to assume something or deny something and the trick is turned. Deny parallels, deny straight lines, deny this postulate, deny that, assume this and assume that, and you have newfangled sciences that can spring up overnight. It certainly requires little originality and less thought;—just nerve and nothing more. Sometimes one feels like laughing, and sometimes one feels a little irritated, that such a hodgepodge could be seriously accepted anywhere for thought.

So far we have an axiomatic geometry that has nothing to do with real things. It is by extension, due to a mere proposition, that it becomes practical geometry, which is not then so much a branch of mathematics as a branch of physics. Geometry thus becomes distinguished into two separate kinds: first, axiomatic geometry, which is pure geometry and pure mathematics; and second, practical geometry, which is a branch of physics. The former treats of pure entia logica having no existence outside of the mind, and hence in this sense mathematics is not a physical science, but a pure logical or formal science.

This view as a philosophic expression of epistemology is of course not Einstein’s own,-and he is not to be blamed for it as his own creation, but it is extremely shallow, and the position of Aristotle and Euclid is in no danger from it. It is too much opposed to common sense ever to obtain even a precarious foothold. Of course such a science as that described by Einstein would not be a science at all, since all science must be a knowledge of real things, and give us a method of exact and ordered knowledge of them. Even he seems to recognize this when he says:

“The investigator in another department of science would not need to envy the mathematician if the laws of mathematics referred to objects of our mere imagination and not to reality.”¹⁰

Thus far we have seen what Einstein thinks of the nature of geometry as a science; we shall now see what he gives as the reason for this stand. Let us hear his own words:

“In a system of reference rotating relatively to an inert system, the laws of disposition of rigid bodies do not correspond to the rules of Euclidean geometry on account of the Lorentz contraction; thus if we admit non-inert systems we must abandon Euclidean geometry.”¹¹

Again:

“Why is the equivalence of a practically rigid body of geometry—which suggests itself so readily—denied by Poincaré and other investigators? Simply because under closer inspection the real solid bodies in nature are not rigid, because their geometrical behaviour, that is, their possibilities of relative disposition, depend upon temperature, external forces, etc. Thus the original, immediate relation between geometry and physical reality appears destroyed, and we feel impelled toward the following more general view, which characterised Poincaré’s standpoint. Geometry (G) predicates nothing about the relations of real things, but only geometry together with the purport (P) of physical laws can do so. Using symbols we may say that only the sum of (G) + (P) is subject to the control of experience. Thus (G) may be chosen arbitrarily, and also parts of (P); all these laws are conventional. All that is necessary to avoid contradictions is to choose the remainder of (P) so that (G) and the whole of (P) are together in accord with experience. Envisaged in this way, axiomatic geometry and the part of natural law which has been given a conventional status appear as epistemologically equivalent.”¹²

We have our old friends with us again, i.e., “if we admit,” “denied by Poincaré,” “assume,” “deny.” Here we have another exposition showing a great absence of logical mysticity, but not mystification or mistiness. Above we heard that all we had to do was to add a proposition, then geometry would become a branch of physics, resting essentially on induction from experience.¹³ Now he makes a partial retraction. Here (P) which represents the proposition above, is allowed to the proposition only in part, “(G) may be chosen arbitrarily, also parts of (P) Then comes the. brilliant conclusion, “the sum of (G)+(P) is subject to the control of experience.” Poor mystic Euclid! You first choose arbitrarily the whole of (G), also parts of (P). Then to avoid contradiction, you choose, choose, mind you, again, whether arbitrarily or not is not stated, the remainder of (P). Einstein does not say why he has to go at choosing (P) in two successive steps. I am afraid it was because of his fear of becoming mystical that he did not explain this.

We should notice the mathematical form that this statement takes. There is a reason for it. We are told first that “practical geometry” consists of two parts (a) “axiomatic geometry,” which is (G); and (b) the purport of physical laws (P). Then (G)+(P) is subject to the control of experience. (G) is chosen arbitrarily, also part of (P); then the rest of (P) is chosen to avoid contradiction. So what we really have is G + xP + yP. Just what each represents is unknown. (G) is known; it is axiomatic geometry; but just what the (P) and its two divisions mean, we do not know. (P) as a whole means the purport of physical law. Yet we can choose part of (P) arbitrarily, and even choose the rest so as to avoid contradiction. But how can we choose to avoid contradiction unless we have a knowledge of that which may be contradicted? If we have a knowledge that depends on the things, how can we choose? But there is no use expecting Einstein to reason.

We are then left completely in the dark as to what relation the purport, or signification, or meaning of physical laws has to (G), which is not related to (P); or how that meaning of (P) can affect (G), except we are told above that it strips it of its formal character. Of this wonderfully clear combination we have only two ideas signalized, “axiomatic geometry” and “purport of physical laws.” Now it is much easier to use symbols than words that have a clear and precise signification. It is easier to explain the unexplainable by means of empty symbols, unknown quantities, than it is to define precisely and exactly in decisive terms what is to be added to what, and how. But it becomes still more confused and awkward, when one of the terms is again divided, to explain the why and the wherefore, as well as the what of the division. Letter symbols are simple if they are not enlightening, and do not leave themselves open to much contradiction since their content is not defined. They afford at least an excellent cover under which to hide an emptiness of ideas. They have, therefore, their reason. But could logic or even nonsense sink to greater bathos.

Coming now to the reason for this distinction, it is simply this, the ideas of geometry are inapplicable to actual bodies because the latter are not rigid, but are in motion. Its final significance, when boiled down and given in simple language, may be put into this statement: “There can be no measurement of extensive quantity because extensive quantity is in movement.” In other words, because there is movement in the physical universe there can be no science of extensive measurement. But if Einstein had stated it thus clearly without the accompaniment of the meaningless jargon we have quoted, the exaggeration and lack of common sense in the statement would be too apparent.

Geometry is the science of static measurement; that is, geometry as a science abstracts from movement and all other conditions, to study one property of matter, viz., extension; the Relativists calmly tell us we cannot study this property because there are other properties of matter, viz., motion. This of course contradicts all notions of science, which consists essentially in reducing complexities to simplicity, by breaking up the concrete multiplicity into its simpler elements, and studying these individually. Geometry is precisely a different science from physics because it studies a different property of matter, viz., extension as abstracted from motion and all other physical qualities; while mechanics is a special science because it studies one special property of matter apart from others, and that is motion. Mechanics abstracts from extension by reducing mass to a point and studying the motion of that point, just as geometry abstracts from motion and studies extension independently of motion. Einstein on the contrary confounds the two, and destroys both geometry and mechanics.

Another distinction that Einstein makes is between “axiomatic” and “practical” geometry; the former is pure mathematics, the latter a branch of physics; the first is no science at all, since it is knowledge of nothing, that is, of nothing real, and therefore of nothing; the geometry that is used in the knowledge of real things, he makes untrue, but he makes it a branch of physics. “Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch of physics. Its affirmations rest essentially on induction from experience, but not on logical inferences only.”¹⁴ This can of course be understood in a correct sense, but the statement as intended is nonsense. Geometry is really a branch of physics. It is that branch that studies the property of bodies called extension, and therefore treats of extended quantity, which is a physical property. This is founded on experience. But that is not the idea contained in Einstein’s explanation. His epistemology is not quite so consistent. His physics is made up of a non-physical science and a non-real one, with a little purport added to make it physical, which purport is a pure assumption, as he tells us. But how a purport can make the non-real real, or how we can choose the purport and at the same time obtain it from experience, we are not told, but are left to make it out for ourselves.

The wonder of this scheme of knowledge is why he singles out that branch of physical science which studies extension or extensive quantity for his peculiar epistemological experiment, apart from the other branches of physical science. If the actual science of quantity is made up of an unreal science plus a certain amount of purport, why should not the same thing hold of the physical science of motion, or of any other property of matter; why should these sciences not be constructed in the same way? What can be the purpose in sequestrating the property of extension from all others for special treatment? It is the same intellect and experience that deals with them all; it is the same physical universe under the aspect of matter and its properties that it studies. If the intellect creates an unreal science of the one, what guarantee or surety have we that it will not do the same with another; or why should it; or rather how could it do otherwise? But logic and consistency are not particularly shining elements in the thinking of Relativists; neither is the explanation of difficulties that their scheme gives rise to.

Now remember that this consideration of geometry is so important for Relativity that without it Relativity is nowhere. Thus Einstein: “I attach special importance to the views of geometry which I have just set forth, because without it I should have been unable to formulate the theory of relativity.”¹⁵ Doubtless; all we can say is that we should all be no worse off had it never been formulated. It would have saved a whole lot of energy and beating about the bush.

But on the other hand it does not bother Einstein one bit to hop to the other side of the fence for a new contradictory combination. The non-correspondence of the science of geometry with physical reality is so important that, without it, Einstein could not have formulated his theory of Relativity. But soon the actual correspondence becomes so necessary that he also needs it in his business. Let us digest this: “Further, as to the objection that there are no really rigid bodies in nature, and that therefore the properties predicated of rigid bodies do not apply to physical reality,—this objection is by no means so radical as might appear from a hasty examination. For it is not a difficult task to determine the physical state of a measuring rod so accurately that its behaviour relatively to other measuring bodies shall be sufficiently free from ambiguity to allow it to be substituted for the ‘rigid’ body.... Not only the practical geometry of Euclid but also its nearest generalization, the practical geometry of Riemann and therewith the general theory of relativity, rest upon this assumption.”¹⁶

Truly the theory of Relativity is wonderful. It rests on the assumption that measuring rods and measurement cannot be applied to reality, for without this view the doctrine of Relativity could not even be formulated. It rests equally upon the absolute contradictory of this assumption that measuring has to be sufficiently free from ambiguity to pass for rigid measurement, and on this assumption “the practical geometry of Riemann, and therewith the general theory of relativity,” both rest. This is the system that is to take the place of the “mystic obscurity” of Euclid. This is a triumph for simplicity and lucidity. The best way to do away with inconsistency and contradiction is to do away with contradiction. If a system so needs a contradictory that it cannot do without it, and needs the opposite contradictory to the same extent, take both; and if you once admit that, there can be no further contradiction; absurdity is killed once for all by being once for all adopted into the home of reason. Anything can go after that, and absurdity becomes even the petted child.

But let us return to the question mooted and see if there is anything to the objection raised by the Relativists. So far Einstein has only given us his view of the science of geometry, and the reasons that led him to adopt this view. He has not yet attacked the question as to whether the practical geometry of space is Euclidean, or Riemannian, or of some other geometric form. It is true we have quoted a statement that the laws of disposition of rigid bodies do not correspond to the rules of Euclidean geometry on account of the Lorentz contraction.¹⁷ There is a double assumption here. First, he assumes a new geometry, which is de facto the Riemannian; and the reason he rejects the Euclidean is that the Fitzgerald-Lorentz contraction is inconsistent with it. The whole question then rests on the assumption of the truth of the Fitzgerald-Lorentz contraction¹⁸ and secondly on the assumption that this contraction, if allowed, is inconsistent with Euclidean geometry.

We are not interested here in the fact of the Fitzgerald-Lorentz contraction, which is a purely physical and rather far-fetched hypothesis; but as Einstein assumes it (assumptions never trouble the Relativists), let us for the sake of argument allow it, and see what would be the result in logic. Even supposing bodies in movement misbehave in the manner supposed, it would not in the least be upsetting to Euclidean geometry. For Euclidean geometry is abstract, and considers-bodies in a static aspect; the only difficulty would be in the application of measurement, and not in the principles applied. This is just one of the peculiarities of Einstein’s reasoning that he talks a great deal about the non-applicability of Euclidean geometry, but never for a single moment does he venture to touch on a single principle of Euclidean geometry and show just where and how it is inapplicable. In fact, he jumps the question entirely after a mere assertion. He never shows himself in the position of a man who has rightly or wrongly reasoned himself out of Euclidean geometry, as did the earlier non-Euclideans; he merely has assumed Riemannian Geometry, and then looks for any reason at all to find an excuse for his method. But his excuses are all beside the question.

We might go further and even say that he dare not treat the topic. For instance, how would the Fitzgerald-Lorentz contraction change the nature of a straight line. Shortening a line does not destroy it. How would it affect parallel lines? Eddington in an example unwittingly admits this: “Let us draw a square ABCD on a sheet of paper, making the sides equal to the best of our knowledge. We have seen that an aviator filing at 161,000 miles a second in the direction AB, would judge that the sides AB and DC had contracted to half their length, so that for him the figure would be an oblong.” The lines are still parallel. The figure is still a parallelogram. Euclidean geometry still applies. Only two sides appear shortened. Shortening either one or both would not make them less parallel. Neither would it affect the sum of the angles of a triangle. At most it would lessen the triangle, or destroy it; it could not change the sum of the angles, if the triangle really existed. But it is easier to assert without proof, or even explanation, if you can find a sufficient number to swallow the assertion.

We might say that the Relativity theory of Einstein touches on what concerns Euclidean geometry in two ways, and in neither does he bring forward anything to contradict, or show impossible, a single truth of Euclidean geometry. These two views correspond to what he calls The Special Theory of Relativity and the General Theory of Relativity. Each might be said to have an entirely different relation to geometry, but neither of them deals with it directly in the least. We shall treat them separately.

Special Theory of Relativity

We shall not expose the Special Theory of Relativity so far as it is a physical theory, for this point of view is of no concern to us in this place. It is only its geometrical implication that we shall take up. The only place where the Special Theory of Relativity comes at all into touch with geometry, is in its attitude towards the coordinate system, or the Cartesian method of analytic geometry. In this system all distances and directions are analytically measured from a stated measuring point, or point of reference, by a system of rectangular coordinates. We shall assume that everyone understands the meaning and method of this system.

Einstein’s attitude towards geometry in the Special Theory concerns only this system of coordinates. He brings this system into mechanics by introducing time, and then he calls it the Galilean system of coordinates. As a preliminary remark, let us add that this introduction of time has nothing to do with geometry; it belongs to mechanics. Yet the whole of Part I of Einstein’s work on Relativity treats of this system, with the exception of the first section on the physical meaning of geometry, and the last on Minkowski’s four-dimensional space. We shall then ask to be excused from quoting directly; it would occupy too much space.

In the first place, what Einstein finds fault with is not the coordinate system as such, whether Cartesian or Galilean, but the transformation formula from one series of coordinates to another. (Transformation of coordinates simply means changing the position of the coordinate axes to which the locus of the equation is referred; this is at times desirable for simplifying the equation. This operation may be represented by a mathematical formula.) Instead of the accepted formula of transformation he substitutes the formula of Lorentz, which is based upon the supposed constancy of light in vacuo with regard to all systems whether and however they may other wise be moving uniformly with regard to the path of the light ray. The difference may be seen in the following equations which represent the so called Galilean and Lorentz transformation:

Galilean	Lorentzian
x’ = x—vt,
y’ = y,	y’ = y;
z’ = z,	z’ = z;
t’ = t,

The only difference between the two systems is that the Galilean system takes into account only the velocity and the time that separate two points of reference from which a moving point is measured, while the Lorentzian transformation formula claims to take into account the time of the light ray traveling from one point to the other. In other words the Galilean transformation formula assumes that the measurements are taken simultaneously; the Lorentzian formula allows for lack of real simultaneity in the apprehending of simultaneity, because of the time taken by light to travel from one point to the other, or the difference in relating the same motion to two points of reference. In addition to this it adds the peculiar hypothesis that light is a physical constant of velocity.

Thus the Special Theory of Relativity discounts simultaneity of events in space by the time taken by light to travel between the point of reference and the objects compared. The simultaneity of subjective perception is therefore supposed to be corrected for the error introduced by the velocity of the propagation of light according to the peculiar Lorentzian hypothesis. Though the velocity of light is around 300,000 kilometers a second, it still falls short of that instantaneous velocity required for perfect simultaneity. All that should be required then would be to allow for deviations from apparent simultaneity, to know the real objective simultaneity to which geometry applies. Even if the Fitzgerald-Lorentz contraction equalizes this difference, Euclidean geometry will apply and will apply just as well as in the other case.

It will then be immediately apparent that the Special Theory of Relativity has no concern at all with Euclidean geometry, and that it is a simple begging of the question for Einstein to say that the Lorentz contraction is inconsistent with the rules of Euclidean geometry. What rules? At the very most he is dealing with analytic geometry, not Euclidean. Euclidean geometry is qualitative, Cartesian is analytic, and finally metrical. This latter does not concern the body of Euclidean geometry in the least. But even in the case assumed, the difficulty does not touch the principles of analytic geometry. It merely changes in a trifling way the formula for transforming one measurement into another. Whether or not there is such a contraction as the Lorentz contraction, and whether or not his formula based on light rays is correct, does not in the least touch a principle of analytic geometry. Whether a distance along a line x is to be x—vt or something slightly different does not have the slightest effect on either Euclidean or Cartesian geometry. To say then that, because of the Lorentz contraction, the rules of Euclidean geometry are inapplicable to reality is mere sophistry. The only difference is in the measuring rod, and whether one measuring rod or another is used formal geometry remains the same.

The transformation formula, whether Galilean or Lorentzian, applies to mechanics, and not to geometry at all. Time and velocity enter into the computations of mechanics, but not into geometry. Einstein is here guilty of a double confusion. He first confuses mechanics with analytical or Cartesian geometry, and then the latter with Euclidean, and finally he applies an argument that has no force even in mechanics.

Thus by making this transformation formula the chief difference between the physical conceptions of Galileo and Newton and his own, Einstein clearly shows that it is not a question of mathematics at all, but one of mechanics. Mechanics deals with motion; geometry abstracts from motion and has nothing to do with it. Whether the mechanics of Galileo and Newton is correct, or whether the mechanics of Einstein is correct, is quite a different question, which we are not interested in—here. But at any rate his case against Euclidean geometry falls fiat, and is not upheld with even a ghost of a reason.

Einstein’s reason is based on a false conception. He assumes that Euclidean practical geometry requires rigid bodies both to be measured and as standards of measurement. Then of course he is forced to deny that there is anything such as a rigid body of reference, a rigid standard of measurement, or a rigid body to be measured.

It is quite evident that there must be some practical rigidity somewhere to measure physical quantity. Nobody, even under the handicap of Euclidean mysticism, ever thought of carrying around a tape measure of a rod or two of streak lightning to measure distance, or a square yard of flowing stream to measure surfaces, or a cubic foot of free hydrogen to measure volume with. One would have to have something to handle that would be fairly steady. But this is not impossible, as everyday experience shows, and as Einstein in one of his better moments clearly recognizes in the passage we have already quoted.¹⁹ If there is any variation in the standard, it is to be taken into account; but that is a matter for practical measurement, not geometry. The case is actual; for example, the standard kept at the International Bureau of Weights and Measures near Paris, is a specially constructed bar of platinum-iridium, and the meter is the distance between two lines on this when at a temperature of melting ice. The possibility of change of standard due to temperature, etc., was recognized, and conditions are realized to keep it as constant as possible; but no one previously ever thought that because a unit of measurement could change by some sort of a movement, all geometry must be thrown overboard, and a new geometry invented. How a new geometry other than Euclidean would be of any help, is also one of the wonders.

Einstein evidently falls into a confusion of ideas. Geometry deals with the static, but has nothing to do with rigidity. Rigidity is a physical quality that does not concern mathematics; mathematics considers one physical property alone, and that is quantity; geometry therefore treats indirectly of the static simply because the consideration of extensive quantity prescinds from motion. Geometry applies as well to fluids in motion as to rigid bodies, but, abstracting from the motion, they are considered in their static aspect, or as simple extension. A static condition is simultaneous everywhere, for since the static abstracts from time, it is equivalent to the instantaneous in time, and as such is comparable.

The old mathematicians and philosophers were perfectly aware of the existence of motion, but they also knew that geometry prescinded from movement, and also from time, except the condition implied in the static sense of simultaneity. Even in fluid and flowing substances instantaneous measurements must absolutely conform to geometry. The only difficulty is in making the measurement instantaneous. But that is a matter for applied mathematics, not for geometry. Einstein rejects simultaneity as meaningless, but here he again falls into a confusion of ideas, for it is not objective simultaneity about which he really argues, but the subjective perception of it, which is quite a different thing. Supposing then, for the sake of argument, that all the reasons he alleges are valid, it would not in the least touch Euclidean geometry, which would still be true, even if the whole Special Theory of Relativity were established.

His second reason is that because of the Lorentz contraction the laws of the disposition of rigid bodies do not correspond to the rules of Euclidean geometry. And he concludes: “Thus if we admit non-inert systems we must abandon Euclidean geometry.” That sounds compelling. Nevertheless, we still admit non-inert systems, and we still refuse to abandon Euclidean geometry. Einstein was not the first to discover non-inert systems as something so new in our experience that we should have to reform our ideas. They have been known since the days of Euclid and before, and still Euclidean geometry throve. In any case the reason given above is a wonderfully powerful argument to effect this change. Einstein must have been assured of its moving power, all by itself, since he did not give it more than two or three lines. Two mere assertions of Einstein, and the thing is complete. To begin with, a system of reference rotating relatively to an inert system has nothing to do with the Special Theory, which concerns only systems in uniform and rectilinear motion. So the whole thing reverts to mere rigidity. Futhermore, rotating reference frames or those in uniform motion have nothing to do with geometry as such, but are, as we saw above, a matter of mechanics; and even if the Lorentzian mechanical theory were true, it would not in the least touch the question of geometry.

The General Theory of Relativity

The General Theory of Relativity comes into even less direct touch with geometry than the Special Theory. It is only negatively that it has the least geometrical concern. It is merely owing to the fact that the equations from which Einstein deduces his physical theory are based on the mathematical equations of Riemann and the work of Minkowski, who made time a fourth coordinate along with the three space coordinates. But it is no longer the Cartesian coordinates that enter in, and there is no longer any real measurement of distance or any of the things for which Cartesian analytical geometry stood.

This merely means that the geometric ideas that are used in the exposition of the General Theory of Relativity, are not those of Euclidean geometry, but those of the non-Euclidean Geometries of Riemann and Minkowski. We shall take up a few of its main ideas separately. But, in the first place, as there is no basis for non-Euclidean Geometry, so there is no basis for any of its apparatus, and the conclusions of analysis built on it can only be as absurd as the propositions from which they set out. Where the foundation is a plain absurdity, the superstructure cannot be real science.

First among the so called non-Euclidean ideas adopted by Einstein is the Riemannian concept of a finite but unlimited universe. Einstein drags this in as a discovery time and again.²⁰ We have already treated this question in speaking of Riemann. Infinite space is not Euclidean but Newtonian. According to Euclid and the philosophy he followed, actual space is limited; so there is no need of a new geometry to give us this conception. It is as old as geometry itself, and we do not need Riemann’s peculiar curved space to arrive at it. It is a much simpler conception.

Another peculiarity of Einstein’s Relativity theory is the assumption of physical qualities for space. This again is an exaggeration of the Newtonian concept of absolute space as something existing by itself. This notion of absolute space was introduced by Newton and further developed by his disciple Clarke. It was shown impossible by Leibnitz. There is, of course, no such thing as absolute space.

Without wishing to enter into a discussion of the metaphysical properties of space, we can at least note the peculiar path of aberration followed by ideas that once leave the normal track. All we know of space is what our experience tells us; we do not know space, but place and extension; that is, without becoming technical, the exact extension of volume to its limits of each material object of our knowledge. The sum total of all these places constitutes real space. But our idea of space is something different. It is no longer the place or sum of places of bodies, but place universalized and abstract. We mentally remove all limits, and all matter, and conceive it as an infinitely extending vacuum. This is of course a pure mental creation with no corresponding thing in reality. It is ideal or conceptual space, that is, possible space as opposed to real space. This possible space Newton considered real, and confounded it with the immensity of God. Then the successors of Newton, led thereto by the wave theory of the transmission of light, filled this space with what they called ether, to account for this transmission. But this ether, though somewhat hazily conceived, was at least a material substance with certain material qualities.

Einstein goes a step farther in making space real by making it something material. He takes Newton’s space, adds to it the conception of the ether, and then to fit his Geometry, brings it back to finiteness, or to the real extent of the physical universe. He thus comes back to the old error condemned by Aristotle in Met. VII, 3, that quantity is a substance, and he adopts the error of Descartes who identifies matter with extension, or space.

In Einstein’s theory the ether, as ether, falls out, and is replaced by a physical conception of space, which then becomes equipped with physical characteristics, and not only performs the function that ether was meant to perform, but becomes the source and manifestation of energy itself.

We can thus see the source of Einstein’s objections to Euclidean geometry, and how they are nothing more than a mere begging of the question. He is only objecting his own hypotheses to Euclid’s science. He invents a special space that has mechanical qualities, and then demands a geometry to treat such properties. Euclidean geometry naturally studies only one thing, the properties of extension, or the possible relation of measurements in three-dimensional space. Einstein supersedes this study by one of mechanical qualities, and then gravely tells us as an objection that Euclidean geometry does not do this. It never pretended to.

Einstein’s view of the ether may be seen in his lecture on Ether and the Theory of Relativity, an address delivered at the University of  Leyden in 1920, as contained in Si(leliyhts on Relativity. He says: “To deny the ether is ultimately to assume that empty space has no physical qualities whatever. The fundamental facts of mechanics do not harmonize with this view.”²¹ We have thus in a nutshell his queer view of geometry; we see how he confounds it with mechanics. It is a geometry of a peculiar kind of material ether. He holds to the ether of the scientist, but his conception is quite different. It is now confounded with space, or rather space-time. After explaining how in his system the metrical qualities of the continuum space-time are the result of the ether itself, he goes on: “But therewith the conception of the ether has again acquired an intelligible content although this content differs widely from that of the ether of the mechanical undulatory theory of light. The ether of the general theory of relativity is a medium which is itself devoid of all mechanical and kinematical qualities, but helps to determine mechanical (and electromagnetic) events.”²²

Here we have another piece quite characteristic of Einstein. He tells us that the ether in his system has now acquired (due to him) “an intelligible content.” Just let us see what intelligible content he puts into it. The intelligible content is that it is devoid of all mechanical or kinematical qualities, but helps to determine mechanical and electromagnetic events. If either he himself or any volunteer (or drafted) commentator can explain the intelligible content of that phrase in an intelligible way, he will deserve well of Einsteinian physics. One thing to be noticed in this, as in so many of Einstein’s definitions and explanations, is that his definition is negative. He never attempts to give any intelligible content, but ever leaves this as hazy as possible. Here he does not tell us what that space of his is; he tells us rather what it is not. It is something that has no mechanical or kinematical qualities. It might not be a whole lot of things, a cow, for instance. It is not any information therefore to tell us what it is not. Yet can anyone inform us what kind of a thing that might be, that has no mechanical or kinematical qualities and yet can produce or help to produce mechanical events?

Such an idea would not be an impossible one, it is true, for the metaphysician. The spiritual is something that has no mechanical qualities, and is conceived as determining mechanical events. But here we are considering pure material things. No doubt Einstein wishes to take mechanical in a restricted sense, and desires to tell us that his space has no mechanical qualities in the sense that it partakes in no way of the nature of force and motion, and that it is, what space has always been conceived to be, pure extension. Evidently he did not wish to define it so, or he would not have made the intelligible content a pure negative indefinite. It only shows again that he does not want to be tied down to precise ideas, because he is unable to precise them.

But let us take this sense; then the other difficulty arises, how can pure extent determine mechanical events? How can space determine without means of determining? How can it give what it has not got? According to all our knowledge of physical things nothing can determine a mechanical event, except that which has mechanical powers. A track can determine the path of a train, a river bed the course of a river, because they oppose resistance. Nothing can determine any motion unless it has something of energy, kinetic or static. But Einstein refuses such to space, and yet he makes it act as if it had. It is only another example of Einstein’s composition of contradictories. He can make a thing have no energy, and yet act as if it had it. He does that by putting together two incompatible ideas, space and energy, and trying to make them one. His plan, instead of separating ideas into their proper species, and genera, and categories, confuses them. Here he fails to discriminate between two separate categories of thought, quantity and action. No wonder he succeeds in making an intellectual monster worse than the horse’s neck and fish’s tail of the Ars Poetica of Horace.

But as usual it does not worry Einstein one bit to take the opposite view. Remember that Einstein’s new ether is nothing but space itself, or rather the space-time of Minkowski. Now about space Einstein thus speaks: “Recapitulating, we may say that according to the General Theory of Relativity space is endowed with physical qualities; in this sense there exists an ether. According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring rods and clocks), nor, therefore, for any space-time intervals in the physical sense. But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.”²³

The ether before had no mechanical or kinematical qualities; now space-time, which as far as we can read his mind (and the same holds for his Relativist followers) is the same as the ether, has physical qualities. It has physical but not mechanical . or kinematical qualities. What are such qualities? He tells us that without them there would be no propagation of light and no space-time measurement. Yet he told us above that his ether differs widely from the ether of the mechanical undulatory theory of light. What physical quality does he put in place of it? He does not tell us; as usual, he just lets us guess. But again he mixes two things, extent, which he terms space-time measurement, and this physical thing that determines the propagation of light. What kind of physical thing can it be, if it is not mechanical? Again we have a negative definition telling us what it is not, leaving us to find out what it is; “It must not be thought of as endowed with the quality characteristic of ponderable media, and consisting of parts that may be tracked through time.” It is a wonder Einstein would not try to forget himself once, and tell us what his ideas are, instead of what they are not. His theory would be a great deal more intelligible.

Now of this whole ether-space-time jumble we have four things told us: (l) it has no mechanical or kinematical qualities; (2) it determines mechanical and electromagnetic events; (3) it has physical qualities; (4) it has not the qualities of ponderable matter. A beautiful double pair of opposites is to be found in this exposition, and never the least attempt to tell us how to combine them together. What physical thing is there that Einstein can describe, or make us understand, that is not mechanical, and yet can produce mechanical effects? That has not the qualities of ponderable matter and yet has physical qualities? If it is not ponderable matter, is it matter at all, and of what kind? It must be matter since it is physical. But what kind of matter can it be if it is not ponderable, i.e., not subject to gravitation, the universal concomitant of all matter, but also not consisting of parts which can be tracked through time? It is thus not only imponderable but immutable. Aristotle’s concept of first matter is vague enough, but at least for Aristotle first matter does not exist as such and is the basis of all mutability. But again it is entirely useless to ask Einstein to give an intelligible explanation of his ideas, or for that matter even attempt it. Much can be forgiven a man who makes a serious attempt to explain what he thinks, but Einstein never gives himself this trouble; just a few vague sentences as above, mostly about what a thing is not, and no more. Such is the Relativist science and thought that are replacing not only mystic Euclid but all the philosophy and physics of the ages. It is sufficient to state what Einstein professes to do, and then observe him doing it for a single page, to know well enough how to class his science properly.

Einstein goes a big step further in objectivizing space than even Newton did. He combines the objective space of Newton, with the ether of physics, and the geometry of Riemann-Minkowski. He himself acknowledges this:“Newton objectivizes space. Since he classes his absolute space together with real things, for him 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, another thing, which is not perceptible, must be looked upon as real, to enable acceleration or rotation to be looked upon as something real.”²⁴ This step Einstein unhesitatingly took, and absolute space becomes the ether with physical qualities.

We have said that Einstein transfers gravitation to the ether: “If we consider the gravitational field and the electromagnetic field from the standpoint of the ether hypothesis, we find a remarkable difference between the two. There can be no space nor any part of space without gravitational potentials; for these confer upon space its metrical qualities, without which it cannot be imagined at all. The existence of the gravitational field is inseparably bound up with the existence of space.”²⁵ The ether or space therefore determines the metrical relations of the space-time continuum. “As to the part which the new ether is to play in the physics of the future we are not yet clear. We know that it determines the metrical relations in the space-time continuum, e.g., the configurative possibilities of solid bodies as well as the gravitational field; but we do not know whether it has an essential share in the structure of the electrical elementary particles constituting matter.”²⁶

It is all very well for Einstein to reduce all geometry to physics by making the gravitational potentials the measure of space, but there is a serious objection that should jump into the view of anyone making such an hypothesis. Even if we grant his assumption that space, or rather space-time, is something physical, something that is measured by the gravitational potentials, and therefore in the order of mechanical force, there still remains the old conception of extension, or even his conception of space-time, if extended, for whose measurement a geometry must be invented. Either the new gravitational space of Einstein has extension, or it is looked upon as a pure dynamic phenomenon without extension. If the first condition is held to be the true one, then space will have a double set of properties, each having a quantitative value, and each will need to have a measure: first, there is the extension of the continuum, and the unit of measurement will have to be an extension unit; then there is the dynamic property of space, and this needs to be measured by a dynamic unit, say the gravitational potentials. We are where we started. Pure geometry is needed for the first. What Einstein has done is merely to take the abstract space of Newton and attribute to it some of the dynamic properties of real matter, and then forget all about the first property, and concentrate his attention solely on the second. It is just the same as if he took a moving body and reduced it geometrically to a point in motion, or a vector, and then made the moving point the whole of the body. He drops extension out of sight altogether. Rather a strange quirk for one putting himself forward as a physical scientist. It would be quite different if he frankly denied extension, as in the second alternative above, and became an out and out dynamist. He would at least be consistent as a physicist, and we should have then to attack him on the grounds of cosmology. As it is, the property of extension is simply juggled out of sight.

Here Einstein not only jumps the question of the measurement of extension, and thus leaves Euclidean geometry out, but takes leave of pure Riemannian geometry as well, since his measure is the physical ether which differs according to him from place to place. The true Riemannian line of constant curvature no longer exists, since it is upset by the local conditions of the ether. Geometry therefore disappears entirely. There is no geometric science, but different physical measurements of a thing physically anisotropic and non-homogeneous. Thus Einstein: “According to this theory (general theory) the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of space and time, or, perhaps, the recognition of the fact that ‘empty space’ in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitational potentials g_µv) has, I think, finally disposed of the view that space is physically empty.²⁷

Einstein thus divides physical reality into two fields, with the hope of ultimately uniting them. These fields we can call space and matter, or gravitational and electromagnetic fields. But let us hear himself: “Since according to our present conception the elementary particles of matter are also, in their essence, nothing else than condensations of the electromagnetic field, our present view of the universe presents two realities which are completely separated from each other conceptually, although connected causally, namely, gravitational ether and electromagnetic field, or—as they might also be called—space and matter.”²⁸

Gravitation has thus become a mere function of space, being nothing more than a manifestation of the properties of space in the physical world. Just one step further would be required to reduce the electromagnetic field to a common denominator, all reality would be reduced to a kind of super-space, and the doctrines of material monism would be completely realized. This result has been envisaged by Einstein: “Of course it would be a great advance if we could succeed in comprehending the gravitational field and the electromagnetic field together in one unified conformation. Then for the first time the epoch of theoretical physics founded by Faraday and Maxwell would reach a satisfactory conclusion. The contrast between ether and matter 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.”²⁹

This condition Einstein professes to have realized in his latest equations, by which he seeks to unite electromagnetism to gravitation. But in doing this he makes a further departure from Riemann’s geometry by reintroducing parallel lines, or what he calls “parallelism at a distance.” A truly wonderful Geometry! When you want to get rid of Euclid, abolish parallel lines; when you need them, reintroduce them. Keep the Geometry that is essentially built on the denial of parallelism, and along with this keep parallelism when it becomes necessary. This is the key to Einstein’s mathematics. He makes the mathematics fit the conclusion, not the facts; there are certainly no limits to Einstein’s ambition, wherever else he is limited. Great indeed is this mathematics, but somewhat in the manner of Diana of the Ephesians; it is all in the shouting. It is wonderful how the whole of natural science and metaphysics can be handled without ideas and notions. It can all come out of mathematical equations—more than was ever put in. Such mathematics indeed would be wonderful. Thus, according to Relativity, all matter, ourselves included, becomes only knots in space-time or rather knots in something of which space-time is the unknotted part; just a manifestation of kinks in the lines and threads of the four-dimensional continuum. This would certainly be a simplification, and in clarity would really be on a par with the present state of geometry, kinematics, and the theory of gravitation, at least as clarified in the Relativity theory. Let us examine some of its principal points as far as Euclidean geometry is concerned.

As we have said before, the mathematics of the General Theory of Relativity does not touch the question of Euclidean geometry at all, except in a negative way, inasmuch as it starts from equations that are not deduced from Euclidean geometry. Einstein follows the conception of Riemann with regard to the nature of the straight line, and his equations of Relativity are built on Riemann’s differential equations of this non-Euclidean line. To that extent he depends on non-Euclidean views of geometry and geometric space, but he has left even this Geometry by making his measure dynamic. In other words, he has created a dynamic Geometry, whereas geometry is essentially adynamic. This non-Euclidean Geometry is merely postulated; not proved, or even assumed for positive reasons.

We were already told by Einstein (but without much assurance) that the Fitzgerald contraction made necessary non-Euclidean Geometry. In the General Theory not only the question of the velocity of light, but the gravitational field, was supposed to make non-Euclidean Geometry necessary. As Einstein himself says: “In view of the results of these considerations we are led to the conviction that according to the general principle of relativity, the space-time continuum cannot be regarded as a Euclidean one.”³⁰ But while in general Einstein bases his theory of Relativity on non-Euclidean Geometry, when speaking directly about it he hesitates as usual, and does not seem so sure. He seems always to be looking for a loophole to escape from a theory on which he has none too firm a hold. “The question whether the construction 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, properly speaking a physical question which must be answered by experience, and not a question of a mere convention to be selected on practical grounds. Riemann’s geometry will be the right thing if the laws of disposition of practically rigid bodies are transformable into the bodies of those of Euclid’s geometry with an exactitude which increases in proportion as the dimensions of the part of space-time under consideration are diminished.”³¹

One thing to be noticed is the last expression of Einstein, for it shows quite characteristically his upside down manner of thought. What he should have said was the opposite; it should be: if the laws of disposition of practically rigid bodies depart from the behaviour of the bodies of Euclidean geometry with an exactitude which corresponds to an increase on the part of space-time under consideration, and approaches the Riemannian behaviour, then the Riemannian is the correct geometry His argument is naturally as follows. Riemannian Geometry will be correct, if Euclidean geometry becomes more and more inaccurate as larger elements of space are measured. He should therefore prove the existence of Riemannian Geometry from the inaccuracy of the Euclidean when extended to the infinitely large. Whereas his proof (?) takes the form of proving Riemannian Geometry not from the fact that Euclid’s geometry does not satisfy the conditions of the infinitely large, but from the fact that it actually satisfies the conditions of ordinary space. He forms an idea in his mind of Euclid’s geometry being the “practical geometry” of ordinary space; Riemann’s of larger portions of space; and then instead of reasoning to this condition from the fact that Euclidean geometry could not be satisfactorily extended to the region where he places Riemannian Geometry, he argues from its filling its own sphere perfectly. He thus begins by assuming the Riemannian conception, and then seeks a proof for it in the fact that it will not apply when the space-time dimensions are diminished, but that Euclidean geometry will then apply. It is at least wonderfully accurate reasoning; it is reasoning that proves nothing at all, except that the reasoner can’t reason. If there is anything to the argument at all, it would be in favor of Euclidean geometry, and not Riemannian. But, as we have said, it is characteristic of his whole method.

With his conception of a non-Euclidean continuum we are not here occupied. It surely is not Euclidean either in its Riemannian or Minkowskian aspect. There is no such thing as curvature of space in Euclidean geometry, and there is no such thing possible as the fourth dimension, and particularly when that dimension is time. Einstein’s theory then is connected with Euclidean geometry only negatively, inasmuch as his assumptions negative all Euclidean geometry. But one can negative anything at all by an assumption. Assumption is not proof, and not even probability. Unless it is some way substantiated, it becomes mere trifling. Furthermore, there can be no demonstrative geometry, Euclidean or otherwise, of more than three dimensions; and for non-Euclidean Geometries there can be none, as we have shown before, beyond two dimensions, within which limits these Geometries correspond to Euclidean geometry of curves. The whole treatment of the question is analytic, and not geometric or demonstrative. Analytically, of course, we can by a mathematical fiction, treat of any number of dimensions, by taking the requisite number of root functions of a quantity. But such analysis is by no means to be transferred to reality, as if mathematical analysis put constraint on reality, and not contrariwise.

Analysis cannot give us more than we can put into it, since all interpretation of analysis is essentially arbitrary and conventional; it depends on the signification we give it. We cannot put into analysis more than our experience allows, and we cannot take more out of it. If we do, we are getting only imaginary conclusions with regard to reality, and this is what Relativism does. Thus, for instance, we can apply analysis to the actual physical coordinates of space. Every coordinate has not only a definite distance, but also a definite direction. We can have an infinite variety of distances, but our directions are limited to the actual directions we can experience. We cannot extend these directions beyond the triple dimension. and anyone who does attempt analysis beyond this is simply deceiving himself with a fiction, and a meaningless fiction at that. Within the triple dimension a demonstrative and qualitative geometry is possible, since we can imagine and even draw the lines, but beyond that we can do neither the one nor the other.

Moreover, what relative length can be assigned to the time ordinate? It is not only physically impossible to adjust it in the three Cartesian planes of Euclidean geometry, but just as metaphysically impossible to conceive it in any sense as a dimension. It is equally impossible to assign to the time ordinate a definite length as well as a definite direction, such as would apply to any fourth coordinate. In what natural manner can a measure of time be made equal to a measure of space? For instance, what is the physical relation between a foot and a second?³² How many of the one are contained in the other? For if there are four dimensions to anything, we must have a common unit to all these dimensions, or again there would only be a misuse of language, and a purposed confusion of thought, or rather making words serve as a cover for no thought.

It is true Einstein would not deal with such a question at all, since he refuses even to consider a univocal measure for either space or time, but places the measure unit in the new continuum, space-time. But the same difficulty occurs here. If they are dimensions, they are dimensional, that is, measurable; and, if measurable, there is a common unit. If no determined unit of length can be assigned to the four coordinates, they are not dimensions in our understanding of the term, and four-dimensional space is only a phrase without background of thought.

Einstein as usual sidesteps this whole question, as he always does in anything that needs explanation. It was for this reason that he built his system on what are called Gaussian coordinates, where every point on a surface is marked by two arbitrary curves, and thus to it are assigned parameters having no actual meaning or measurement.³³ In this system coordinates have no physical significance, and simply serve to determine the points in the continuum in an arbitrary manner. In other words, the General Relativity Theory is an attempt to express physical laws independently of any and all frames of reference. But expressing law independently of dimensions is one thing; assuming extra dimensions, and denying dimensional measurement, is quite another. It is easy to assume when no account has to be made for the assumption.

The differential equations of the line in Einstein’s calculation have no coordinate significance, and the coordinate is only of importance analytically for the summation in manufacturing the equation. The gravitation potentials have now become the measure, and the element of Riemann that before was a line element, is no longer such but is now really a gravitation element. Einstein has therefore completely assumed and analyzed himself out of geometry. He is in physics pure and simple. Physical as is the so called Riemannian geometry, Einstein has gone even beyond that. His measurements have not the slightest connection with space or extension measurements; they are mere force units, made from the potentials of gravitation. Nor is his space even Riemannian. It is now not only not isotropic, but not even homogeneous. His gravitation potentials differ from place to place, and so consequently does the nature of his space. They are different in the neighborhood of matter and away from it; and different near one center of matter and near another. The Riemannian curve and line must, therefore, be abandoned as well as that of Euclid.

Whether Einstein has a correct physical theory or not, he certainly has nothing in the world to do with geometry. Were Einstein’s theory true a thousand times, we should still have Euclidean geometry, and his theory could not be true where it contradicts that geometry; unless indeed we accept antecedently, or merely assume after the manner of Einstein, that all physical measurement such as we have always known it, is a snare and a delusion, and extension itself a non-entity. But then what would become of Einstein’s curves in space-time? Even if we grant for the sake of argument that the gravitational and electromagnetic fields act as his equations suppose them to act, Euclidean geometry would still be true as long as there is a science of measurement of extension.

As long as there exist extension and dimensional quantity, there must be a dimensive measure and dimensive relations, and a dimensive measure that is univocal with the thing to be measured. We cannot have a bushel or a yard of weight, and we cannot have a gravitational unit of distance. Einstein has again merely fooled himself with words. He read somewhere (it is in Riemann, but it is really as old as the hills) that the measure of continuous quantity must be something outside itself. The meaning of this is that in a continuum there is no natural unit, as there is in discrete quantity, and accordingly the unit of measure is arbitrarily imposed from the outside. But outside does not mean outside the sense in which it is to be measured; we cannot take a pound of distance, a thing which Einstein really tries to do when he makes gravitation a measure of space. A measure of space must be a space measure, the unit must be univocal, or of the same nature as the thing measured by the unit. A gravitation unit can only measure gravitation, or something in the same order, such as a force. It cannot measure extension or space. Since Einstein’s space is neither the real nor notional space of measurement, but something material in itself, if this thing have extension, it must also be measured quantitatively as space. We would therefore have to supply a new space measure for his hyperspace, and we should again have Euclidean geometry.as before. Or, if it is not extended, his equations become meaningless. He is again caught on the horns of a dilemma.

Clearly, then, if we accept Einstein’s position, we shall have done away, not only with Euclidean geometry, but all geometry. For him it cannot exist. What Einsteinism really does is to destroy all science of measurement, and all measurement itself. He has been talking about geometry, but he was using empty words. What he had in mind was his own peculiar system of mechanics, by which he intends to replace all physics as well as all geometry by mere analysis. This is what his system comes to and it is on this count it must be mathematically judged. Anyone holding that force iB a measure of distance is really obliged to deny extension and space, and hence space measurement, and all mathematics as well, since there is no longer really dimensive quantity. In that case he should have to sweat to explain the meaning of a mathematical curve in space, or even in the mythical time-space. Evidently Relativity has given us a pretty kettle of fish, but they are all pollywogs. Einstein will doubtless have a problem on his hands to make humanity abandon its ideas for his, once it grasps what the latter are about.

Einstein has in this proceeded much further than even his premises allow. We saw before how Einstein’s notions of the inapplicability of Euclidean geometry and the necessity for some form of Riemannian, arose from his connecting the idea of rigidity and non-rigidity with geometry. He imagined that Euclidean geometry required for its application absolute physical rigidity. The absence of this quality requires, according to him, the giving up of measurement and dimension.

A favorite representation of the Relativists is to compare the four-dimensional continuum to a kind of jelly through which the world lines, or geodesics, run, on which are marked “events” of time-space. Now this medium, by no means rigid, may be distorted into any shape, yet the world lines always run through it in the same order. Time-space may be so considered; each observer’s time-space is distorted relatively to the time-space of another, just as the jelly may now be in one shape, now in another.

This supposed property has even given rise to a peculiar branch of mathematics called Analysis Situs worked up chiefly by Riemann, Betti, and Poincaré. It is a mathematical study of a continuum, where magnitudes and directions are not always measurable because of changing conditions. Analysis Situs is therefore that branch of analysis (falsely called geometry) that studies the positional relations of the different elements of a figure, irrespective of distance and direction. Its propositions remain true no matter how the continuum may be distorted; for the positional relation of the points remains in some sense the same no matter how distance and direction may be changed, since the continuum contains certain properties apart from distance and direction. This Analysis Situs is of course preeminently a non-Euclidean study, with no value in practical science.

In accordance with this notion Relativists reduce the term dimension to that of order. Length and duration are supposed to have no counterparts in the external world, and there exists only a certain ordering. Thus in the jelly the medium may be distorted anyway, but the order of events on the world lines and the Analysis Situs remain the same. Let us hear Eddington on this. He is one of the clearest and most logical: “Although length and duration have no exact counterparts in the external world, it is clear that there is a certain ordering of things and events outside of us that we must now find more appropriate terms to describe. The order of events is of four-fold order; we can arrange them as right-and-left, backwards-and-forwards, up-and-down, sooner-and-later. . . . It is recognized at once that there is no essential distinction between right-and-left and backwards-and forwards. The observer has merely to turn through a right angle and the two are interchanged.... The amalgamation of up-and-down is less simple. There are obvious reasons for considering this dimension of the world as fundamentally distinct from the other two. Yet it would have been a great stumbling-block to science if the mind had refused to combine space into a three-dimensional whole. The combination has not concealed the real distinction of horizontal and vertical but has enabled us to understand more clearly its nature—for what phenomenon it is relevant, and for what irrelevant. . . . We must now go further and amalgamate the fourth order, sooner-and-later. This is still harder for the mind.”³⁴

Notice the peculiarity of the reasoning here; it shows how far a false theory will mislead even a clever man. Eddington tries to distinguish fundamentally between up-and-down, backwards-and-forwards, and right-and-left. He says that all you have to do is to turn through a right angle to have the latter two interchange. But all he has to do is to turn through a right angle to find either of the latter two interchange with the first. Let him lie on his bed and get up on his feet, and he has accomplished it. There is not the slightest possible distinction thinkable between up-and-down, backward-and-forward, and right-and-left as dimensions. The only distinction is one unconnected with dimension; it is one of gravity; and that is between down and up themselves and not between up-and-down and any other pair. We fall down, not up; it may be either right or left, backward or forward. But of course the whole thing was merely plotted to give some plausibility to the introduction of the other pair necessary for the four-dimensional continuum—sooner-and-later.

We have said that Einstein went further than his logic allowed. If his theory were true, there would be room for an Analysis Situs of the various positions of events, and the relation between these might possibly be expressed by gravitational potentials, since Analysis Situs prescinds from direction and distance. But then there could be no four-dimensional absolute interval. The moment an absolute interval enters in, quantitative geometry comes back, and quantitative measurement also.

Even in Analysis Situs distance cannot be done away with. The four-fold order of events requires distance and direction. The distortion cannot destroy them, or the four-fold order ceases to be. How can there be forwards or backwards, right or left, up or down, unless there are distance and direction between the extremes. If there is no distance between forward or backward, there is no forward and backward, and there is even no order. Analysis Situs simply considers the order, abstracting from the distance and direction, but these latter are there, and the moment we make any interval absolute, they return and demand measurement. Gravitation measurement therefore will never do as a measure of extension, or dimensive quantity, even if the continuum is a fantastic jelly-like space. It would still require an extensive measure.

Of course we are simply deducing the necessary conclusions from Einstein’s theory, and not attempting to explain his actual thought. That, as we have so many times urged, is practically impossible. Einstein has such a faculty for embracing both sides of a contradiction that one would have to be of the same frame of mind to follow his thought, it is so peculiarly his own. His thought is but odds and ends, unconnected bits, incongruous, undigested, and contradictory. Whatever he is as a pure mathematician dealing in pure mathematical symbols, he becomes the most out-and-out careless thinker the moment he gets beyond his symbols and his equations. He is guilty of the very vice he attributes to Euclid, that of spinning a web of the pure logical-formal, with no relation to reality; and then when he tries to establish contact between this and reality, his thought staggers, and reels, and stumbles, and falls, like a blind man rushing into unknown territory.

Although we are interested in Einstein’s peculiar developments of his physical theories, only in so far as they touch our present subject, geometry, nevertheless his strange geometric assumptions have certain physical implications that are, to say the least, most extraordinary and fanciful. In the first place, one curious thing is that he sees no opposition between claiming Relativity as a science of greater universality and truth than that of Galileo and Newton, and on the other hand he never thinks of applying it universally. Einstein would limit his theory to the physics of the vaster portions of space, and would leave the world as we know it to Euclidean geometry, Cartesian analysis, and Newtonian physics. But principles refuse to limit themselves according to the whim of their promulgator. If his theory is true, it should replace all geometry and physics; if it cannot do this, it is not a general theory at all; at most it can only be a particular case.

If a four-dimensional space governs physics in interstellar and interplanetary distances, then it should govern events on this planet of ours. Since the Relativity theory unifies inertia and gravitation by bringing them to the same thing, all mechanics should be reduced to the mathematics of gravitation. It is all the more necessary, therefore, that Einstein’s theory take over the responsibility for the whole of mechanics. Because, while we do not consider gravitation in many physical problems, mass and inertia are in all of them, and since inertia and gravitation are unified, they should have the same treatment.

If time is a coordinate elsewhere in the universe, it also a coordinate in terrestrial mechanics. If the gravitational potentials constitute the measuring relationship t all, the relationship is universal. There can be no reason for restricting these relations, except one of pure contriving to avoid bringing the system into touch with accepted fact for fear of its being contradicted, or of lack of assurance that it would apply, or of inability to make it apply. It is not merely a case of the perihelion of Mercury or the bending of a ray of light when passing through the field of the sun, or the displacement of the spectral lines towards the infrared in light coming from the stars. It should embrace all nature, if it is really a universal theory. It is certain that heretofore there has bee no effort made to apply Relativity to general mechanics.

This is all the more true because, if the theory of General Relativity is true, ordinary mechanics is not. For instance, if the idea of the curved line is correct, and the simplest path, or geodesic, of a body is a gravitational curve, then the principles of mechanics are wrong. The laws of motion would be incorrect. They suppose the possibility of motion in a straight line. In mechanics a; curved paths are due to more than one force operating in different directions. If any curved path is a simple geometrical and mechanical element, then all mechanical laws of motion as used and formulated are wrong and should be replaced by a law that represents reality It is useless to say that practically there is not an appreciable difference in small distances between the straight line and the curved. It is not at all a question of quantity of difference, but of difference itself; it concerns truth and falsehood, not mere expediency; it is a question of principle. If in any curved path there is only one force acting, then two forces acting in different directions are not necessary to produce a curved path, and any theory making such a claim a fundamental postulate in its system is erroneous.

Again, if time is anywhere a fourth dimension for world events, it is everywhere. Time cannot be a different measure in physics in various places. If Einstein’s conception of a four-dimensional finite but unlimited continuum is a true conception of the whole world, then every part of the universe is but a real portion of this continuum. The four-dimensional measurement cannot be applied to one part of it, and an entirely different set of measurements to another part. Time cannot be an essential part of the whole continuum, while at the same time it is an entirely separate element in one part of the whole, measuring simple change in a three-dimensional continuum.

If Einstein is correct in his system, all mechanical formulae built on time as a separate measurement are false and misleading. They should be thrown out of mechanics. That such demands should be made of a General Theory, never even seems to occur to Einstein and the Relativists. They see no contradiction in advancing a theory that denies validity to existing theories, and leaves the better part of the field exclusively to these theories. It only gives us another view of the combination of contradictories everywhere to be met with in Relativity. Relativists not only want to have their own cake and eat it, but they deny the other fellow the existence of his cake, and still want to eat it.

This brings us to another beautiful consequence of trying to hold contradictories about the same thing. If the universe, as Einstein holds, is a four-dimensional continuum, then there is no such thing as succession of time in a three-dimensional continuum. Every event in the world is synchronous just as all parts of a continuum are co-existing at the same time. The four-dimensional continuum is either actual or it is not. If it is not, there is no such thing as an actual four-dimensional universe. If it is actual, then there is no such thing as succession in the universe; it is altogether at once, just as two portions of space co-exist together, as New York, London, Paris, Rome. This is recognized by Eddington, the well-known English exponent of Einstein’s Relativity. He says: “Events do not happen, they are just there, and we come across them.”³⁵

In such a case everything would have to be really simultaneous, and change is only a situation in a different spot of the space-time continuum. This would apply universally as well to terrestrial as to astral physics. For instance, it would be impossible for a falling body to change its position in a three-dimensional continuum in successive moments of time, since there is no such thing. The whole change would be a mere static condition of the wonderful space-time. The notion of succession would have to be dropped out of time, and therefore all actual change, even of successive positions in space-time, is impossible.

Just to show the peculiar quirks in the Einstein mental make-up, let us see another example. Einstein goes to great pains to show there is no such thing as simultaneity in the world. Of course, as usual, he is beside the point. All he really attempts to show is that we cannot perceive simultaneous things, because of the time taken by light, the messenger of simultaneity, to come from simultaneous events to the reference point of simultaneity. That his argument does not even show this, we shall let pass for the moment. But, even if this were proved, it would not show that there is no simultaneity. For this is a question not of perception of simultaneity, but of existence of simultaneity. Rome and New York coexist together, though it may take me ten days to become personally cognizant of their co-existence. Let us allow the whole parallogism, just to get at the conclusion. What Einstein is so intent about, in finding difficulties for mechanics, becomes a bomb to explode his own system. If things are not simultaneous then by all means there is succession and change in the world, and there is time as a separate dimension.

But now just when he has proved as against Newton that succession must be so rabidly held to that there is no possibility of simultaneity for the scientist, he does away at one stroke of his pen with all succession for his theory, and postulates absolute simultaneity. The contradictory action implied in the proverb, “It depends on whose ox is gored,” is the acme of consistency compared with this reasoning. For, if time is a mere static dimension, there can be no succession in the universe; no time in the sense of mechanics and physics where time is the measure of change of position in a continuum of three dimensions. Yet, on the other hand, Einstein says there is no simultaneity. Perhaps what is static in four dimensions is kinetic in three. But the world is not built at the same time on two different plans, with two different sets of laws. Einstein is certainly a wonderful creator of chimeras, but instead of a lion’s head, they have a bull’s head, for they have two horns, and they always impale their creator on one or both.

There are some other implications contained in this wonderful theory. For instance, we never change, we are not really born, nor do we die; we are simply spread out over a portion of space-time like some huge sprawling monster. This is recognized by Eddington, who at least is not afraid to deduce the conclusions from Einstein’s Relativity. “An individual is a four-dimensional object of greatly elongated form; in ordinary language we say that he has considerable extension in time, and insignificant extension in space. Practically he is represented by a line—his track through the world.”³⁶ Just try to get a mental photograph of yourself to see what you look like when you have a figure of the three ordinary dimensions and one other dimension of three hundred thousand kilometers for every second of your existence.

Eternity, too, is reduced to a very simple thing. We are at least as changeless and eternal as the four-dimensional world of which we form a portion. The poet’s conception of time that never returns becomes false, for it never goes away. No part of time flows, any more than New York flows, or London flows, or Peking flows. It simply remains where it is. The old metaphysical religious questions that have long troubled the world are all solved. We are immortal; eternity is with us, for we never change. Since all the events of our life are but simple positions in space-time, they are there as they always were. Our consciousness is just playing us a prank, when it tells us of change and of growing old; and our intellects deceive us when they represent life and death.

Minkowski too holds to a static condition of these points in the time-space continuum. “The substance of any world point may always with the appropriate determination of space-time, be looked upon as at rest.”³⁷ For if every world point is at rest, the sum total of the world points, or the world of space-time would be at rest without change or motion. This would necessarily follow from making t a coordinate corresponding to the space coordinates, x, y, z. Thus Minkowski declares: “Let the variables dx, dy, and dz, of the space coordinates of this substantial point (real world point) correspond to time element dt, then we obtain an image of the everlasting career of the substantial point, a curve in the world, a world line, the points of which can be referred to the parameter t from — infinity to + infinity.³⁸ The world line corresponding to such elements of four coordinates is a static curve in the four-dimensional continuum, just as is any curve corresponding to an analytic equation in three-dimensional geometry. Each substantial point is everlasting and unchanging, and so are all together.

But of course Einstein does not mean any such thing as a four-dimensional space with a time coordinate univocal with the space coordinate. He does not mean it, although he tells us he does. “Formally, these four coordinates correspond exactly to the space coordinates in Euclidean geometry.”³⁹ “Under these conditions the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time coordinate plays exactly the same role as the space coordinates.”⁴⁰ “According to the theory of relativity, the ’time’ x ₄ enters into natural laws in the same form as the space coordinates, x_l, x₂, x₃.”⁴¹ He does not mean it though he repeatedly calls his “world” a “space-time continuum.” He does not mean it although his arguments and language necessarily demand it. For the fourth dimension must be static as are the other three, otherwise it would not be a fourth dimension exactly as the other three are, and to call it so would be using words without meaning. Einstein does not mean this because he is still obsessed with the idea of motion, and apparently motion in a four-dimensional continuum. It is only another case of Einstein wanting to have his cake and eat it. He wants to eat up time by turning it into a static fourth dimension; he wants to keep it by still keeping motion of which time is but the measure. Aristotle defines time as the number (measure) of movement, estimated according to its before and after.⁴² Like space itself time has three dimensions, the moving present (nunc flues), the past and the future. The past is no more, and the future is not yet. The only reality is the moving now. It is this motion of time that makes change possible and at the same time measures it. It is this nunc fluens that makes it entirely different from the static dimensions of space. It requires a different kind of measure from that of space. It is not a measure of extension at all, but of movement. Einstein with great volubility and argument classes it as formally the same as the dimensions of space. Then immediately he forgets it, because he never understood what he was saying, for he still keeps to movement in his space-time world. It is a wonder it would never occur to him that since he has placed our three-dimensional time among the extension dimensions, he would have to have a new time or supertime to measure movement in his four-dimensional space-time. Whichever way he turns he is again caught on one of the contradictory horns of a dilemma.

Motion, mutability, and time are correlatives, just as are immutability and eternity. If time is in the same order as a dimension of space and a mere fourth dimension in a space-time continuum, and the whole continuum is a simultaneously existing limited curved sphere, there can be no mutability, no motion. The whole of the fourth dimension must co-exist with the other three. Motion along the world threads is impossible; there is no place for change with time as its measure. If on the other hand there is motion, that which moves along the world threads is something different from the threads themselves, and cannot be measured by the thread measure, the gravitational potentials. If there is motion, there must be a measure of the motion, for change necessarily means a different condition before and after. This only can be time and space. The four-dimensional measure, the gravitation potentials, cannot measure this, for the measure of what is static is static itself, and the gravitation potentials only show a configuration of the world threads. The measure of motion must be in the same category as motion itself, i.e., time and space.

We are then back in full dualism of measure relations if we grant any movement in four-dimensional space. Unless it is a pure static monism, the monism of Relativity will throttle itself. Unless all motion is suppressed in its physics, we have the absurdity and patent contradiction of making time on the one hand a dimension of the static continuum, and on the other of reintroducing it as a special measure of change. For, if anything moves in this continuum, we must have a measure of the motion; in other words, reintroduce time, or invent a new time that will be a univocal measure of change. To put the whole matter in a nutshell, either the Relativists have to deny all motion, or after all their running around, they have to come back to just where they left, to time as an essential measure of motion.

But let us again concede for the sake of argument the conception which seems to be that of Relativity, of a four-dimensional continuum in which there is movement. The implications are no less astounding than before. Since the world is both finite and limitless, its dimensions are so likewise. We have only to keep in movement to come to the same position in space-time from which we set out; we only need to complete the cycle to have all return again.⁴³ The threads of time and space are limited by the extent of the world, and since they are curved, they return whence they came. Eternity is therefore simply time returning endlessly on itself, a simple following of the world-thread around the unending circle. We shall live again; we shall have all the events of our life again. Life will be periodic; the same things will occur again when the great circle of the world thread shall have been completed. Again, each life has assigned to it a definite portion of the world thread, which is not discontinuous but a continuum. There is then no room for individual existence; birth, life, death, are nothing but points or positions on the world line, and they merge into one another as in a continuum. There is no natural division or separation. If the world of space-time is a sphere as is supposed, the only distinction between different careers is that between different world threads, but on the same world thread all is a continuum. As time must be in one direction, all careers must return whence they came; everything is in cycles; and the whole universe will be periodic.

The whole Relativity theory is a running around in confused circles, with no reason or logic, and is as easy to follow as the path of a bat in the air at night. The Relativists either mean what they say when they reduce time to a fourth dimension, or they are simply talking in the air. If they really mean it, they should deny change and motion, and all such things they should banish from physics and biology as pure deceit. If they mean it, and still preserve motion, they are committing the absurdity of holding both sides of a contradictory at the same time. If they do not mean it, but merely take time in the sense in which it was used by Lagrange, as a mathematical function of the three dimensions in estimating kinetic problems, then the whole incoherent mixture of Relativity is but a smoke-screen to camouflage the absence of thought within.

This is only another example of lack of clearness of conception and love of the incongruous that everywhere characterizes the position of Einstein. The question arises, how can we explain such a confused mentality in anyone professing himself to be a scientist? The explanation seems to lie in the fact that Einstein is neither a physicist nor a metaphysician capable of dealing clearly with physical and metaphysical entities in a scientific way. He is merely an analytical mathematician seeking to give a physical interpretation to the conclusions of his mathematical process. In this he is hampered by a load of contradictory and absurd assumptions of the school that he follows, which throws him into all manner of difficulty. But above all he utterly lacks the scientific sagacity or instinct to choose the proper physical, metaphysical, and mathematical basis for the working out of his process; and as a result his beginnings are as bad as his conclusions. But even worse than this is the absence of the power of criticism to enable him to see the glaring contradictories which he is embracing, and the lack of logical insight to understand the use and the force of language. No wonder then he blunders, and in floundering, grasps for odds and ends left him by his predecessors in the same wild school of thought.

Take the difficulty in the whole question we have just now been treating. How could anyone, if he stopped to consider a moment, mix up two such diverse intellectual concepts as time and extensive quantity? One answer that may be made is that it was done for him already by Minkowski. But the gist of the matter lies in the lack of perception or penetration that should have shown him that every measure must be the same in kind as the thing measured; that a measure of the static cannot be a measure of motion; that change and immutability cannot have the same kind of measure. He could not see that a measure of movement must be something in motion, i.e., time; the measure of an extent, an extent itself. Otherwise he never would have jumbled an extent unit and a motion unit into a univocal measure, and then by an illogical mathematical twist have dropped them all for another measure that is neither extension nor motion, but force; and finally, he would never have reduced force again to a curvature, which is a curvature of a non-dimensional continuum of no real extension. It is certainly a funny jumble in measures.

He is far, therefore, from being the kind of innovating scientist that was, for instance, Galileo or Newton, whom he criticizes; and his conception of the mechanics of the universe is far removed from that set forth, right or wrong, in the Principia. While Newton too was a mathematician, he was also a physicist, and whether or not we accept his physics in their entirety, we must admit that he could give a great deal of consistency and logic and plausibility to what he thought on the subject. Relativity, to say nothing of its truth or falsehood as a scientific theory, but considered only as a presentation of an hypothesis, is certainly not at all in the same class as being the “Principia” of a new science, as was the work of Newton in his day.

Relativity can impose itself only in one way, by imposition, by advertising, or whatever you will, on those who are willing to concede that something awfully profound and recondite must be hidden in the thick haze of big words and incomprehensible ideas, or who are willing to accept at its face value the claim that Relativity is something so far beyond their comprehension that they have nothing to do but accept the dictates of its prophets without question.

Another thing that strikes one in the expositions of Einstein is the struggle he engages in to show that his ideas are not inconceivable. Thus his term, space-time, which is really meaningless to the human intellect (it is nothing but a term, for what he represents by it is as inconceivable as that of the other compound—square-circle), he strives to explain, and even suggests that he can clarify it, but not all of his followers follow him here. Many⁴⁴ of them frankly confess that the ideas of Relativity are inconceivable, and that we should not try to conceive them; but not Einstein. And here, again, we see the strange kink in the mind of Einstein, as strange as the kinks he puts into the world-threads to constitute gravitation in space. In seeking to explain the inconceivable parts of his system, he exhibits the most solicitous care in explaining at length what is clear and, one might say, evident. Einstein is not always the writer of vague and nebulous language, and impossible conceptions, that we have shown. At times he can be as clear as any writer. He can be lucidity itself when he is explaining the trite commonplaces of science that all accept; it is only when he starts to explain what should be his own, that his language becomes confusion and his thought disorder.

He exhibits great care in explaining what really needs no explanation at all, while omitting all mention of just that which should be explained. Evidently he insists only on that which he can himself understand. A rather curious example of this is seen in his essay on Geometry and Experience in “Sidelights.” Let us read what he says: “Can we picture to ourselves a three dimensional universe which is finite yet unbounded?

“The usual answer to this question is ‘No,’ but that is not the right answer. The purpose of the following remarks is to show that the answer should be ‘Yes.’ I want to show that without any extraordinary difficulty, we can illustrate the theory of a finite universe by means of a mental image to which, with some practice, we shall soon grow accustomed.”⁴⁵ He then begins by defining what he understands by “visualizing.” “First of all, an observation of epistemological nature. A geometrical-physical theory as such is incapable of being directly pictured, being merely a system of concepts. But these concepts serve the purpose of bringing a multiplicity of real or imaginary sensory experiences into connection in the mind. To ‘visualize’ a theory, or bring it home to one’s mind, therefore means to give a representation to that abundance of experiences for which the theory supplies the schematic arrangement.”⁴⁶ In this he gives us his theory of knowledge and its relation to reality. But as we are not occupied with Einstein’s theory of epistemology, we shall let it pass.

He then goes on to show how to “visualize” the answer he proposes to give. “In the present case we have to ask ourselves how we can represent that relation of solid bodies with respect to their reciprocal disposition (contact) which corresponds to the theory of a finite universe. . . . So, will the initiated please pardon me, if part of what I shall bring forward has long been known.”⁴⁷ In what follows he might ask pardon of the veriest tyro of a schoolboy for taking all the pains he does to explain what is as plain as any pikestaff. The only thing in it is that his profuse explanation gives him a chance to clutter up the question with confused ideas. He first explains what is meant by an infinite continuum. He shows that any number of equal solid bodies can be placed side by side without filling infinite space. Then he goes on to an infinite continuum in a plane, where the solid bodies are replaced by squares of cardboard. From this he goes on to illustrate what he calls the two-dimensional continuum which is finite but unbounded. He takes as example the sphere, and shows that we can move a paper disc round and round without arriving at a limit or boundary; but that the number of discs that can be laid side by side on the sphere is limited. And remember all this explanation of a supposedly profound problem was made to the Prussian Academy of Sciences. It only goes to show how little it takes to make an audience think it heard something.

While Einstein’s theory of epistemology does not interest us, its practical application does, particularly the extraordinary steps he takes to make us visualize a three-dimensional space finite yet unbounded, and most especially the brilliant mathematical and physical reasoning by which he leads up to it. It is so characteristic of the Einsteinian mind. Einstein starts out to show us the finite by means of the infinite. He seems to suppose we are some kind of transcendent intellectual creatures who are quite familiar with the idea of the infinite, but who only by an effort and “some practice” can be led by the hand and brought down to grasp the finite. Thus he gives us two examples of the infinite, an infinite volume and an infinite plane to make us realize what is meant by the finite but unbounded surface of a sphere. I fear ordinary teachers do not sufficiently realize this advance in epistemology, and that every schoolboy should be made to comprehend the manner of finiteness of his baseball and football by means of his much more accurate knowledge of infinite planes and volumes.

Nor is this all. We find again reproduced Riemann’s marvelous distinction between limitless and infinite. The latter belongs to the measure relation, the former to the extent relations. Einstein falls into the same puerile blunder, like one blind man following another and falling into the same pit. They both confuse the meaning of the words by exchanging the one for the other, as far as the significations can be distinguished. A thing is really or objectively infinite when it actually has no bounds. It is subjectively infinite or unbounded, or as far as we are concerned, when we cannot measure to the end of it. Now what do Riemann and Einstein do? They turn the meaning around, and make infinite that which we cannot measure; and therefore infinite and finite become mere subjective terms. A finite surface is one on which we can complete this operation.

But the real stroke of genius is to follow. It is contained in the two different meanings of limitless or unbounded. The juggle of changing the definitions gives Einstein a chance so to cloud the meaning of words that most readers will not know what it is about, and will be ready for the further equivocation in the meaning of limitless or unbounded. Unbounded as referred to the extent relations of the universe means exactly what negative or possible infinite means; there is no limit or bound to the extent. We cannot put a bound, and say the extent will stop there. It can be extended infinitely or indefinitely in the negative sense. But what have we got here in Einstein; is it this sense of limitless? Not at all. We have an entirely new sense for limit and bound and hence for limitless and unbounded. Bound or limit now simply means a fence, barrier, or other obstacle to physical motion. Limitless merely means a place that may be run round in or on with nothing to stop the running.

A sphere is limitless if you can push a disc on it round and round without obstruction. A backyard is finite and limitless if there is room enough to run around in it. A cow pasture would be finite and limitless if the cattle followed the fence around instead of running into it. There is a rule that says a man may define his terms as he wishes, provided he sticks to his definition. While I should normally object to such an unwarranted use of the word limitless, I should not grudge it to Einstein, for, if he should stick to it, it would not avail him much for his theory of Relativity. A limitless universe that need not be any more limitless than a boy’s baseball ought to cause no commotion.

Now with regard to the term limitless and the term boundless: in actual fact Einstein has not shown us the infinite at all; his argument shows us only the limitless, and the measurable and immeasurable, which have nothing at all to do with infinity. The gross fallacy lies in transferring the meaning from the first case which signifies limitless, or potentially infinite, to the silly meaning that would apply to a baseball or football. Such is the twaddle that has set the scientific world by the ears concerning a finite yet limitless universe. Faithful believers must have thought it meant something.

But let us go on further. The only limitless thing in the whole argument seems to be absurdity. Einstein has been juggling before our eyes the infinite, the finite, and the limitless, and here is the conclusion from the juggle. After trying to place the round flat discs on a spherical surface and finding that they do not fit as on a plane surface, he adds this astounding conclusion: “In this way creatures which cannot leave the spherical surface and cannot even peep out of the spherical surface into three-dimensional space, might discover, merely by experimenting with discs, that their two-dimensional space is not Euclidean but spherical space.”⁴⁸ Here again we have the old catch of Fechner and Helmholtz concerning the experience of two-dimensional beings. But let that pass. If those beings were Relativists, we might expect anything; but if we suppose them to be reasonable and logical, they should discover just what we know, that they were on a spherical surface.

The whole argument centers round the fallacy that a spherical surface is a two-dimensional surface. Einstein repeats this again and again. Now there is no such thing as a two-dimensional spherical surface. There is a two-dimensional projected surface of a sphere; there is a two-dimensional developed surface of a sphere; but there is absolutely no two-dimensional spherical surface. There is not even a two-dimensional curved surface, such as a cylindrical surface, though on a cylindrical surface the discs might fit very well. The reason is quite plain. Every curved surface is essentially three-dimensional, that is, its directions necessarily bring in the three dimensions; by joining its extremities we always obtain a volume. In other words, any portion of a spherical surface can hold water, which Einstein’s argument cannot; for the spherical surface is always the superficies of a volume.

The triple dimension has to be used to measure the surface itself. And thus it is absolutely distinct from the plane surface of Euclidean geometry. Any reasonable being on such a surface, even if he were confined to the surface, would realize by any respectable system of measurement that he was moving on the surface of a solid sphere, just as we know the same with regard to the earth, without having to be beings that can penetrate it as empty space.

Here Einstein and the other formulators of this and kindred arguments, are victims of their own extravagant nominalism. They are so cocksure that their science deals only with concepts and terms with no relation to reality, and that that relation is forced on reality by a decree of their own, that they cannot recognize the difference in a term when it has different significations. They create the term surface, as they think, and when in accordance with their theory they endeavor to add the purport, or the relation to real things, they find it will not fit. But instead of criticizing their definition to make it fit with reality, they squeeze reality to make it fit with their definition. They invent a real absurdity rather than relinquish their right to create arbitrary science and arbitrary concepts.

But for people who are not obsessed by a shallow and irrational epistemology, and who try to keep their concepts in touch with real things, surface does not by any means always signify the same thing. We even signify this when we make a specific difference between plane surfaces and spherical surfaces. This means that surface is only a generic term, and that therefore we cannot reason from it to the various species without distinction, any more than because cow and man are both species of the genus animal, we can conclude that man walks on four feet because cows do. This is really the implicit argument of Einstein in the application of the term surface in this question. A plane surface is two-dimensional, a curved surface three. It would be just as sensible to argue that the superficies of a hexahedron or a dodecahedron is measured in two dimensions, as it would to use that argument for the surface of a sphere. In the one case each gives two dimensions, and the total of directions marks out a volume. It is the same with a sphere, except that the portions measured two-dimensionally must be smaller; they are infinitesimals. The portion ds2 is all that can be measured in two dimensions; a neighboring ds will already give a new direction theoretically sufficient to deduce the triple dimension. The fundamental directions are just as manifest in the tangents to the different points of the spherical surface as they are in the tangents to the surfaces of a dodecahedron. The whole difficulty comes from a confusion of the idea of plane which is two-dimensional, and surface, the idea of which is not representative of dimensions at all, but of the term, or limit, or superficies of a body.

Suppose, and this would be a real supposition, that the beings described by Einstein lived on a real plane surface of two dimensions, then we might say they would have no experience of the third dimension; but on a spherical surface it is quite different. The spherical surface really requires three-dimensional space to be realized, and if the beings concerned could not find that out it would be the fault of their intelligence, not of the conditions of their existence. Let us then grant that the inhabitants of Einstein’s sphere could never get beyond the notion of a curved line; that they could never measure more than in two directions, and therefore that they would have no sense of three dimensions. The three dimensions would exist nevertheless, and that is the important point, and not the peculiar psychology of Einstein’s imaginary people.

The upshot of the whole argument is to be found in the claim that a curved surface cannot be made congruent with a plane surface without change. A cylindrical surface can be made congruent with a plane surface by bending only; a spherical surface must be bent and also stretched to make it congruent with a plane surface. But the reason of this is precisely because they are not two-dimensional surfaces, but tridimensional, and the third dimension has to be corrected before it can be made to coincide with the two-dimensional surface. Contrariwise also, plane circular discs will not fit on a curved surface, for they are made for fitting in two-dimensional space, and not in three-dimensional space. The whole argument from surfaces falls flat, for it is not a question of whether surfaces as such have length, breadth, and thickness, that is, whether they themselves represent volumes, but whether or not they show three-dimensional space. They do, and hence a curved surface necessarily implicates three-dimensional space.

Such is the explanation of the so-called two-dimensional curved space. Thence Einstein argues to show the possibility of three-dimensional curved space. He adds: “Now this is the place where the reader’s imagination boggles. ‘Nobody can imagine this thing,’ he cries indignantly. ‘It can be said, but cannot be thought. I can represent to myself a spherical surface well enough, but nothing analogous to it in three dimensions.'⁴⁹ But patience, dear reader, you have actually imagined it; your spherical surface was already pictured to you in three dimensions, and therefore you cannot represent something else different as being also curved in three dimensions. You are just being fooled with words; keep your common-sense visualizations; they give you the truth. The one who is talking to you of a three-dimensional curvature different from the ordinary spherical surface does not know what he is talking about.

Let us observe Einstein do some more juggling. We may listen for amusement if for nothing else. “We must try to surmount this barrier,” he tells us, “in the mind, and the patient reader will see that it is by no means a difficult task.”⁵⁰ Our author does not seem to think it a difficult matter at all to show us how to see this three-dimensional curved surface, and to surmount the barrier. But let us see how he is going to make us represent this newfangled curved space that was never seen on land or sea or in the air.

“For this purpose we will first give our attention once more to the geometry of two-dimensional spherical surfaces. In the adjoining figure let K be the spherical surface, touched at S by the plane, E, which, for facility of presentation, is shown in the drawing as a bounded surface. Now let us imagine that at the point N of the

spherical surface diametrically opposite to S, there is a luminous point, throwing a shadow L’ of the disc L upon the plane E. Every point on the sphere has its shadow on the plane. If the disc on the sphere K is moved, its shadow L’ on the plane E also moves. When the disc L is at S, it almost exactly coincides with its shadow. If it moves on the spherical surface away from S upwards, the disc shadow L’ on the plane also moves away from S on the plane outwards, growing bigger and bigger. As the disc L approaches the luminous point N, the shadow moves off to infinity, and becomes infinitely great.”⁵¹

Here we have again our old friend two-dimensional spherical surface, but Einstein immediately substitutes in its place an honest-to-goodness sphere; a mere surface could not act, as he has to make it act; but that is a small detail for Relativity. Even then neither sphere nor spherical surface has the slightest thing to do with the illustration. Neither one is required for his purpose; all he needs is a hole in the wall and a light behind it; or just as simple, a plain round disc with a light behind it. The disc then casts a shadow, and as the disc is moved relatively to the light, the shadow becomes larger and larger, and approaches infinity. But when he is talking about the geometry of spherical surfaces, he has to use the word at least in the beginning.

Here is how he explains that three-dimensional spherical geometry is representable. All we have to do is to imagine spheres that can be brought to coincide with one another, and behave like the disc shadows, i.e., increase in radius according as they move further out, “then we shall have a vivid representation of three-dimensional spherical space, or rather of three-dimensional spherical geometry.”⁵² We are afraid we shall have to confess to seeing nothing of the kind either vividly or clearly. We could imagine all kinds of such spheres, but it would not change our notion of three-dimensional space one bit. We fear it is but another bit of suggestion to make people believe that they can see it. Too much like hocus-pocus.

Of course Einstein imagines the lines drawn from surface to surface as the spheres increase in size, to be the real measures of distance, and these lines would be curved. But what help does that bring his scheme? At most he has only curved lines, and they can be imagined and produced without all the rigmarole he has gone through. This conception does not change the nature of space dimension one bit; the line connecting the centers of these spheres will still be straight, and will give the absolute distance.

In this argument we have the fundamental nonsense of the whole Riemannian and Relativistic concept of space. Curved space is supposed to be something different from curved surfaces and curved lines. The argument is primarily based on the fallacy of arguing from a two-dimensional curve to a three-dimensional and four-dimensional curved space, just as we proceed from a two-dimensional plane to a three-dimensional volume. That is the whole argument; let us look at it again. First, Einstein talks of surfaces. Mind you, a two-dimensional surface that has no physical existence apart from a volume. Then he turns it into a sphere such that every point on the sphere casts its shadow on the plane. He is therefore in plain three-dimensional space and is not dealing with surfaces at all. This is the most absolute sophistry. What he should be talking about, and what his system requires, is curvature. What he should do, if he had any argument to present, is to show how curvature applies to lines, surfaces, and space. But that would be to be logical, and to be logical would mean to get stuck. He prefers a method of argument that will throw dust in the eyes. Hence he argues from two-dimensional surfaces to three-dimensional curvature; whereas he should have argued from two-dimensional to three-dimensional curves. Now every circle or rather every circumference, i.e., of the disc, is a two-dimensional curve, and every spherical curvature is a three-dimensional curve, as we have shown. Einstein prefers to deceive himself and hoodwink others by calling the sphere two-dimensional, and then trying to make us imagine something three-dimensional, when the relation of the spherical surface to space is already three-dimensional.

The three-dimensional relation of curvature to space is then the spherical surface, and there can be no other three-dimensional curvature of space. It is then pure nonsense to say that probably “our three-dimensional space is also approximately spherical, that is, the laws of disposition of rigid bodies in it are not given by Euclidean Geometry, but approximately by spherical geometry, if only we consider parts of space which are sufficiently great.” This spherical curvature in three dimensions would have to coincide with the spherical surface, or, in other words, the whole of space would be cut into spherical surfaces. Now this is very simple, if we care to represent a sphere as a continuum of spherical circles enlarging from the center out; but this is no contradiction of Euclidean geometry, but plain everyday Euclidean geometry with no hitch in it. Every such sphere has a radius and a diameter, and thus ordinary geometry holds. There is nothing new here.

This is where Einstein supposes the reader’s imagination to boggle. “Nobody can imagine this thing,” he says. “It can be said, but cannot be thought.”⁵³ It is here he introduces the disc experiment as a confirmation. But how about the argument? After creating disc shadows on the plane that increase in size according as they move farther on the plane from the disc, Einstein concludes: “The shadow geometry on the plane agrees with the disc geometry on the sphere.”⁵⁴ How about it? All about it is this: disc circumference corresponds to disc circumference in shape, and that is all. There is nothing three-dimensional in it. A plane f3at disc with light behind it would do the same thing. All the talk of the spherical surface and curves in two dimensions and three dimensions is again pure mental sleight of hand. Einstein shows one thing and uses another. He shows a three-dimensional surface, even a three-dimensional sphere, but he uses only a planar disc.

Now comes the great conclusion of the whole thimblerigging. “The representation given above of spherical geometry on the plane is important for us because it readily allows itself to be transferred to the three dimensional case.”⁵⁵ But how? Here we shall have to revert to the language of Einstein himself, for his argument is too ineffable to be subsumed in any other words. “Let us imagine a point S of our space, and a great number of small spheres, L’ , which can all be brought to coincide with one another. But these spheres are not to be rigid in the sense of Euclidean geometry; their radius is to increase (in the sense of Euclidean geometry) when they are moved away from S toward infinity, and this increase is to take place in exact accordance with the same law as applies to the increase of the radii of the disc shadows L’ on the plane.

“After having gained a vivid mental image of the geometrical behaviour of our L’ spheres, let us assume that in our space there are no rigid bodies at all in the sense of Euclidean geometry, but only bodies having the behaviour of our L’ spheres. Then we shall have a vivid representation of three-dimensional spherical space, or, rather of three-dimensional spherical geometry.”⁵⁶

That is all there is to it. But great is the power of suggestion. We are simply told we have it. We must confess to being an unfit subject for this species of hypnotism. We can certainly represent to ourself clearly, how vividly we need not explain, the imaginary behaviour of the small spheres; but we have come no nearer to a conception of three-dimensional spherical space, though we might assert that we had a fair representation of three-dimensional spherical geometry, which for us simply means the Euclidean geometry of the sphere. Before analyzing the argument we wish to call attention to one thing. Einstein set out to make us picture to ourselves a three-dimensional universe which was finite yet unbounded. That was on page 45; here we are on page 55. After ten pages of beating about the bush, showing the evident, and cluttering the rest up with mistakes, we have our whole showdown in the above paragraph. This is an illustration of our assertion that Einstein is very solicitous about explaining what is obvious, but the real point that demands explanation he slides over, and does not see it at all. After telling us something that we need no great stretching of phantasy to imagine, we are simply told that we have our vivid imagination of three-dimensional space. It is precisely this connection which should be established, if Einstein is arguing and not suggesting. But it is useless to argue. Let us see if there can be a connection.

Just as in the case of the disc shadows on the plane, so the increasing spheres in space show us nothing in the world but a wonderful phenomenon of expansion that follows a regular law. The fact that the spheres are multiplied, does not change the geometry or the space; two spheres or ten spheres or ten million spheres have the same geometry and the same space arrangements as one sphere; neither does the fact of their increasing physically in size. It is hard to see how a greater or less can change the application of geometry, and certainly Einstein does not tell us how.

There is only one more of the data given that we have not mentioned, and it is that this increase of size as the spheres depart from S is not to be detected by means of a measuring rod, because the measuring rod will behave in the same way as the spheres. In that case as far as we are concerned the increase in size might as well never take place, since it will never be perceptible to us; and the only conclusion we can draw is that our calculations of the depth of space will be erroneous, or space will be much greater than our measurement of it. But as that part of space to which we can apply measuring rods is rather small, it will not make much difference. But if it affects all things equivalently, it will not make any change either in our geometry or in our physics, further than that our calculations of interplanetary and interstellar distances would not be objectively true, but at least they would be subjectively the same for everybody. All we could say is that nature had contrived to deceive us in this respect. But neither spherical, Riemannian, nor any other geometry could correct it for us, or represent it to us, if for no other reason than that we cannot represent what we know nothing about.

With another clear deduction that “space is then finite, because in consequence of the ‘growth’ of the spheres only a finite number of them can find room in space, “⁵⁷ which of course is false, Einstein closes his famous argument on Geometry and Experience. It is false unless the spheres themselves become infinite. Then. if the spheres are infinite, one of them will fill infinite space. This is only important as showing that Einstein’s reasoning remains with Einstein to the last. He seems to want to make sure that he will not have a single argument correct.

The last argument we considered in illustration of his peculiar idea of space may be taken as a typical example. There is of course no argument in the whole eleven pages of the exposition; not even clear illustration; a great deal of the plain and obvious, with here and there an obscuring of the question by false assumption, and everywhere sophistical argument, as if it were contrived especially to keep one from perceiving clearly what the writer is driving at. For instance: the introduction of the spherical surface, then a sphere, finally the disc whose shadow projection was studied for the conclusion. Let me show you some geometry connected with two-dimensional surfaces. Just look at this sphere. Now watch how this tiny disc acts when there is a light behind it, and it or the light moves. Now that is spherical three-dimensional geometry, if you can imagine solid surfaces acting like the shadow of the disc. Pure mental legerdemain! Euclid, the mystic, and Einstein, the magician! Even if there had to be a choice between the two, science at least should be able to choose.

Plenty of inaccuracies also are furnished for the brew. Examples: every point on the sphere has its shadow on the plane. Not true; at most a hemisphere can cast its shadow on the plane. Moreover the shadow of the disc has nothing to do with spherical surfaces. It is the circular boundary that counts. What is inside the boundary can have any shape; it will be all the same. So the spherical surface we set out with does not enter into the illustration at all, except as hocus-pocus words. The finiteness of the plane is proved by the fact that only a finite number of shadows can be cast. Wrong again. The plane is potentially infinite because there is no limit to the number of such shadows. So the argument or illustration of a finite world falls. There is further a wrong conception of the infinite and of the unlimited; of the dimensions of the curve; there is the introduction of the idea of rigidity, as if it were a condition of Euclidean geometry. But this ought to be enough to show what value should be attached to Einstein’s views on geometry, or to his criticisms of Euclid’s mysticism. 

References

1. Sidelights, Geometry and Experience, p. 28.   
2. Relativity. Sect. I. pp.1-2.   
3. L.c., p 3.   
4. Sidelights, p. 30.   
5. L.c., p. 31.   
6. Relativity, p. 3.   
7. Sidelights, p. 30.   
8. Sidelights, p. 32.   
9. L.c., p. 33.   
10. Sidelights, p. 27.  
11. L.c., p. 33.   
12. L.c., p. 35.   
13. L.c., p. 32.   
14. L.c., p. 32.   
15. L.c., p. 33.   
16. L.c., p. 37.   
17. L.c., p. 33.   
18. The Fitzgerald-Lorentz contraction was offered as an explanation of the Michelson-Morley experiment. This experiment was undertaken to show the effect of the “ether drift” on the passage of a ray of light through it, which should take place if the earth were moving at a certain velocity through the ether. The result of this experiment way entirely negative, whereas there should have been a positive result, if the earth were really moving through an immobile fluid ether. Fitzgerald of Dublin suggested that the thing that held the mirror used in the experiment shrinks when it is moving. H. A. Lorentz of Leyden carried this further and asserted that the electric theory of matter required that such a shrinkage should occur. The Fitzgerald-Lorentz contraction is therefore a supposed shrinkage in length of a moving body in the direction of its motion. 
19. Sidelights, pp. 36, 37.   
20. Cf. Relativity, Sect. 31, The Possibility of a Finite and Yet Unbounded Universe, p. 128; Sidelights. p. 41.   
21. Sidelights, p. 16.   
22. L.c., p. 19.   
23. Sidelights, p. 23.   
24. L.c., p. 17.   
25. L.c., p. 21.   
26. L.c., p. 20.   
27. L.c., p. 18.   
28. Sidelights, p. 25.   
29. Ibid.   
30. Relativity, p. 111.   
31. Sidelights, p. 39.   
32. Relativity actually makes the speed of light, three hundred thousand kilometers a second, equal to one second of time. But of this later. 
33. Einstein actually does away with all measurement in the accepted sense. He first transfers his system to Gaussian coordinates, and then these disappear and leave only a result in the terms of gravitation potentials. Gravitation is therefore the measure, and the measure of gravitation reverts to a unit of curvature. A dynamical unit turned into a unit of the adynamical. 
34. Eddington, Space, Time, and Gravitation, Chap. II, p. 35.   
35. Space, Time, and Gravitation, Chap. III, p. 21.   
36. Space, Time, and Gravitation, Chap. III, p. 57.   
37. The Principle of Relativity, by E. A. Lorentz, A. Einstein, H. Minkowski and A. Weil. Space and Time, by H. Minkowski, p. 80, tr. by W. Perrett and G. B. Jeffery, New York, 1923. 
38. L.c., p. 76.   
39. Relativity, p. 68.   
40. Relativity, p. 67.   
41. L.c., p. 147.   
42. Phys. IV, 11.   
43. Some of Einstein’s followers are not afraid to deduce the logical consequences of the theory. G. N. Lewis of the University of California makes all the phenomena of the physical universe reversible in space-time. Eddington also allows cyclic reversion, S. T. G., Ch. X, pp. 161. 
44. Cf. Poincaré: “Anyone who devoted his life to it might perhaps in the end form some idea of the fourth dimension” ; also, cf. Schrödinger World’s Work. June. 1929. 
45. P. 45.   
46. Ibid.   
47. L.c., p. 46.   
48. Sidelights, p. 50.   
49. Sidelights, p. 50.   
50. Sidelights, p. 50.   
51. L.c., pp. 51, 52.   
52. Sidelights, p. 55.   
53. Sidelights, p. 50.   
54. L.c., p. 52.   
55. Sidelights, p. 54.   
56. L.c., p. 55.   
57. Sidelights, p. 55.