CHAPTER IIIHistory of Non-Euclidean Geometry
On this there are two things to be remarked, and these two things will show the immense distance that separates the clear and limpid intellect of the ancient Greeks from the foggier intelligence of the moderns. In the first place, these arguments, though they contained many surprises, never got beyond the controversial stage. The arguments might be clever, surprisingly clever; they might even to some extent be unanswerable. But they never for a minute stopped the clear flow of Greek thought, or imposed themselves on the Greek mind as based on reality. They were only clever controversies to bring out the truth, no more. The Greek intellect (intus legere) was more penetrating and clear than was its reasoning power; so that even when the reason could not detect and answer the fallacy, it knew it was there, and was therefore calm and undisturbed by such argumentation. Among the moderns we find the opposite attitude. They prefer to believe and assent rather than to controvert. A fallacy is proposed and immediately becomes the basis of a school of thought. It requires only one individual to start a falsehood going, without even an attempted proof, to find immediately plenty of following. Another distinction is that the ancients at least tried to prove the opposite contention. The moderns do not go that far. They merely assume or deny. None of the neo-Geometers or their followers ever thought of furnishing a proof for the contradictory of Euclid’s postulate, although they are loud in demanding apodictic proof for it; they merely deny the one and affirm the other. Such is the method adopted for the trumped up and loudly trumpeted non-Euclidean Mathematics of the moderns. Still to hear them, traditional mathematics is but a beginning as compared with theirs. Here is an example of one of these ancient arguments against the meeting of the lines. Proclus, before giving his own proof, examines one of them. The argument is as follows. If AB and CD be two converging lines forming with the line AC interior angles less than two right angles, then by measuring in the following way they can never meet. Bisect AC at E, and along AB and CD take AF and CG equal to AE. Join F and G. Repeat this operation for FG and each successive base as for AC and FG.
Then AB and CD can never meet on any of these bases, for if they did, two sides of a triangle would be equal to the third. For if AF and CG met on FG, then AF plus CG would be equal to AE plus EC by construction; and so on for each succeeding measure. Proclus, indeed, was not able to expose the fallacy, which is one of the same kind as the race between Achilles and the tortoise. It is at least much cleverer than anything urged by the neo-Geometers in favor of their postulates, but it is none the less a fallacy. It is simply an argument based on the infinitely divisible, and it would tend to show the lines as pure asymptotes, which would perfectly satisfy Lobatschewsky and Bolyai. The term of this variable is the point of intersection which the variable approaches but does not completely attain. But there is no such limit for the actual lines. They not only converge but intersect and then diverge. Their intersection is not on any such cross line as those given but beyond it. We thus see that non-Euclidean Geometry is not a brilliant conception that first sprung up in the brains of two or three nineteenth century mathematicians, but goes back almost to the foundations of scientific geometry itself. The moderns only added the necessary temerity to stake their all on the non-Euclidean opposite, without knowing any more about it and argumentatively perhaps much less, than the author of the fallacy examined by Proclus. The next step we find represented by Saccheri, whom moderns acclaim as their precursor. He is the precursor in the sense that he begins to let his controversial attitude run away with him, when, instead of merely pointing out the weakness of the argument in the proof of Euclid’s postulate, he urges the possibility of the contradictory and cultivates a real methodic doubt. We have seen an instance of this attitude in his criticism of the argument of Nasiraddin at-Tusi. Nasiraddin supposes that with converging lines a perpendicular to one will make acute angles on one side and obtuse on the other continuously as the perpendiculars become shorter or longer. Saccheri objects to such an assumption. His objection however is not to the lack of proof of the assumption of the Persian geometer, an objection he was clearly entitled to make, but his objection takes the form of the question, “Why should not the angles on the side where they are acute become greater and greater till finally we arrive at a right angle, and a transversal perpendicular to both lines?” His question brings us to an inconceivable position, as if that were the point at issue. Lambert is much more logical and reasonable. Speaking of the status quaestionis he says: “The question itself concerns neither the truth nor the conceivability of the Euclidean postulate.”1 He does not put the Euclidean statement on a par with an inconceivable opposite. It is this type of criticism that the neo-Geometers so loudly acclaim in Saccheri, and it is the same as that on which their own systems rely. Saccheri’s own proofs as far as they concern the acute and obtuse angles, are of the same kind, and it is for this reason and no other that he is held up to honor by the non-Euclideans. But in all fairness, the old Greek sophist who invented the argument examined by Proclus should be put on the same pedestal, if not indeed above him. He at least tried to give the theory a positive foundation. Its modern authors are satisfied with the mere negative basis of a denial. The fact that no satisfactory demonstration had ever been found for the parallel postulate of Euclid thus became the occasion and the sole reason for the launching of a new Geometry, not only not depending on this assumption, which would have been entirely legitimate, but based on one absolutely incompatible with it. What Proclus had foreseen as possible, that the imagination might be led in the opposite direction, had occurred. We have already seen the growth of this scepticism. In the short treatise of Kaestner and Klügel, which we mentioned before, we saw doubt expressed as to the possibility of demonstrating the parallel theory; but they considered that it would be absurd to attack it, and that it should be held because of experience and ocular evidence. But scepticism soon advanced much farther than this. The doubt of the possibility of a proof soon degenerated into a certainty of its impossibility among many mathematicians. Gauss was one of the first to doubt its demonstrability and to foresee the possibility of a Geometry based on contrary principles. Its indemonstrability has been considered proved by Beltrami in 1868, and by Hoüel in 1870.2 In fact at the present time it seems to be accepted as an axiom that the postulate is incapable of proof. The better known mathematicians of today all hold this. Heath in a note to the discussion of the fifth postulate simply states that the indemonstrability has been proved by Beltrami and Hoüel. Poincare states: “The postulate of Euclid then cannot be demonstrated; and this impossibility is as absolutely certain as any mathematical truth whatsoever.”3 Progress in certitude with regard to the non-Euclidean position has certainly been great during the last century, and mathematical minds have surely made a great leap from mere doubt in the beginning, or even from the state of mind of Lobatschewsky, one of the founders of non-Euclidean Geometry: “The futility of the efforts which have been made since Euclid’s time during the lapse of two thousand years, awoke in me the suspicion that the ideas employed might not contain the truth sought to be demonstrated”; or from the frame of mind of Bolyai, the other founder, who later in his life, in studying the matter further, thought at one time he had discovered a proof of the Euclidean hypothesis, and was willing to admit that by the aid of solid geometry evidence might be obtained against the consistency of non-Euclidean Geometry.4 There is now no longer a mere suspicion of doubt; the truth of non-Euclidean Geometry is proclaimed as if it were the last word in certitude. The first of whom we have positive record as doubting seriously of the truth of Euclidean geometry and the possibility of another geometry not only independent of the fifth postulate but opposed to it, was Carl Friedrich Gauss (1777-1855), who was looked upon as the leading mathematician of his time, “princeps mathematicorum.” He occupied himself with this theory for many years, in fact, according to his own confession, during the whole of his adult life-time. He recognized that all the attempts to prove the fifth postulate and fill in the gap that existed in Euclidean geometry had been futile, and that his own researches amounted to no more. He became convinced that geometry could not be established as an a priori science,5 and that the gap in Euclid could never be filled. On the other hand he held that a consistent geometry could be built up while refusing to accept the truth of this axiom. This he called anti-Euclidean or non-Euclidean Geometry, and the latter name stuck. His view of geometry was that it was a consistent building once the parallel axiom was granted, although he made a further objection to the definition of the plane, which he considered to involve a theorem. The position of Euclid was however capable of near proof by experience, but if one refused to grant this, a self-subsisting Geometry could be devised without it and with the contrary principle. Gauss published nothing of this during his life-time, because, as he says, he was afraid of the clamor of the Boeotians (da ich das Geschrei der Boeoter scheue)6 and possibly also, on the other hand, held by sane doubts of its truth. He was certainly not confident enough to risk his reputation on the new Geometry, and he had to see others take the palm from him. His stand only became known when, after his death, his own letters and those of his friends were published. That he had some intention of writing in the sense of non-Euclidean Geometry is clear from his letter to Schumacher,7 but nothing was found among his manuscripts; so either he failed to follow up his intentions, or else destroyed his efforts. Another that worked on the same problem as Gauss and with the same convictions was Ferdinand Karl Schweikart (1780-1857). Gauss was also acquainted with his work, for in a letter to Schumacher in 1846,8 he speaks of a certain Schweikart who called the new Geometry Astralgeometrie. Schweikart also published nothing of his investigations in this line. The only thing he published of a mathematical nature was in 1807, his Die Theorie der Parallellinien nebst dem Vorschlage ihrer Verbannung aus der Geometrie. But here the autor, despite the title, had no notion of a new Geometry, but of a strictly Euclidean one, which he wishes to establish. Later he engaged in investigations along the lines of Saccheri and Lambert, and this led him to the development of a non-Euclidean Geometry. But that is about all we know of his efforts. His nephew, Franz Adolph Taurinus (1794–1874) whose attention had been called to this question by his uncle, has published his own investigations on the subject, Theorie der Parallellinien in 1875, and Geometriae Prima elementa in 1826. Taurinus, however, was still convinced of the truth of Euclidean geometry, and his aim was to succeed where Saccheri and Lambert had failed, i.e., to prove the third hypothesis contradictory, that of the acute angle. But in doing so he approaches non-Euclidean Geometry as the Euclideans Saccheri and Lambert did, and Lobatschewsky and Bolyai in a frankly non-Euclidean sense. He came to the conclusion that besides Euclidean geometry, there was another which he called logarithmic-spherical and which he considered non-contradictory, and he developed to a certain extent the trigonometry of such a system. He also admitted the possibility of a third Geometry, that of the sphere, such as was afterwards adopted by Riemann. But the real founders of non-Euclidean Geometry were Lobatschewsky and Bolyai. It would seem that the question was in the air, and that several were tentatively circling about it. But youth rushed in where age with more prudence feared to enter, and the prize, if it was one, was seized by the more adventurous. There was not really much left them in the way of discovery; the path had already been shown by the later investigators of the Euclidean postulate, particularly by Saccheri, and perhaps by Lambert, and the new founders were doubtless influenced by their thought. All that was required was a little more boldness, to throw over all connection with the old, and the new Geometry was in large part already to hand. This boldness was found in a young Russian professor, and a young Hungarian army officer. Nicolaus Ivanovitch Lobatschewsky (1793-1856) was a professor of mathematics in the University of Kasan in Russia. He began by working on the theory of parallels with the idea of proving it. Regarding this he says. “A rigorous proof of this truth has not hitherto been discovered; those which have been given can only be called explanations, and do not deserve to be considered mathematical proofs in the full sense.”9 But before 1829 he had changed his mind, for at that time he published The New Basic Principles of Geometry with a Complete Theory of Parallels. In 1826, he placed before the University of Kasan a treatise never published, entitled: Exposition succinte des principes de la Géométrie avec une démonstration rigoureuse du théorème des parallèles. It was therefore somewhere in the three years between 1826 and 1829 that he became converted from one certainty to another, from belief that he had solved the riddle for Euclid, to belief in an entirely new and contradictory Geometry—rather quick work in terminating the investigation of ages. In the last work mentioned above he showed the possibility of another Geometry independent of the parallel axiom. He wrote several works on this new Geometry, among them being Imaginary Geometry in 1835. He was a prolific writer and continued to the end of his life to write concerning the new Geometry. In 1855 a digest of all his researches, called Pangeometry, was published in both Russian and French. All his writings are now edited in his Collected Works. John Bolyai (1802-1860) was a Hungarian officer in the armies of Austro-Hungary. He became interested in the parallel theory through his father Wolfgang Bolyai, from Bolya in Siebenbürgen, who was much interested in mathematical questions. By 1823, his son had built up a body of Geometry wherein the postulate of Euclid was replaced by one incompatible with it. The work containing this development was published in 1832 as an appendix to a work of his father’s, Tentamen juventutem studiosam in elementa matheseos purae, elementaris ac suhlimioris, methodo intuitiva, evidentiaque huic propria, introducendi. Cum appendice triplici. The third appendix contained the work of his son, a treatise of only twenty-eight pages, with the title, Appendix scientiam spatii absolute veram exhibens: a veritate aut falsitate Axiomatis XI Euclidei (a priori haud unquam decidenda) independentem; adjecta ad causam falsitatis, quadratura circuli geometrica. Auctore Joanne Bolyai de eadem, Geometrarum in Exercitu Caesareo Regio Austriaco Castrensiurn Capitaneo. He retired from the army in 1833, but published nothing further, though it seems he occupied himself in investigating and extending the principles of the new Geometry. This new Geometry of Bolyai and Lobatschewsky did not gain much recognition at first, and attacks on Euclid’s postulate were looked on as acts of irresponsible and muddled heads. Both died disappointed, and one certainly embittered. But after their death the allusions of Gauss to their work, and his own leanings in that direction, as published in his letters after his death, again drew attention to it. The writings of both Lobatschewsky and Bolyai had been brought to the notice of Gauss, the work of the latter by his father, an old university acquaintance, and he found in them ideas on which he himself was working. His favorable references to the new Geometry did much to establish it among mathematicians. Translations were soon made into various languages, especially by Hoüel into French in 1866. Others also were made in German and Italian. Both works have been translated into English by George Bruce Halstead, Chicago, 1891. The germ of the new Geometry was in the peculiar notion of parallel lines with which it started. According to Euclid’s geometry only one line can be drawn through a point parallel to a given line. In the new Geometry an infinite number of such lines can be drawn, all of which will be non-secant This of course is based on the assumption that the angles of a triangle are less than two right angles, which is Saccheri’s third hypothesis, the one which was the most unsatisfactory as far as the establishing of the Euclidean principle was concerned. Bolyai’s notion of parallel lines is substantially the same as that of Lobatschewsky, but is expressed in a somewhat different way. For Bolyai parallel lines mean what they mean in ordinary geometry, i.e., lines that are equidistant. There are other lines according to him that are non-secant, or will not intersect no matter how far they are produced although they are not equidistant, and these he calls asymptotic. For Lobatschewsky there is no such distinction, they are all parallels. According to him all straight lines in a plane which pass through a certain point can be divided into two classes with reference to a given straight line, viz., those which cut the line and those which do not. The line which forms the boundary between these two classes is called parallel to the given line.
The angle formed by this line and a perpendicular on the given line is called the angle of parallelism, or angle DAF in the figure. If this angle of parallelism is a right angle, Lobatschewsky shows that the sum of the angles of every triangle would have to be equal to two right angles, and Euclidean geometry would follow. But if this angle is an acute angle, then the sum of the angles of a triangle will be less than two right angles and non Euclidean Geometry is the true geometry, which Lobatschewsky assumes and calls Imaginary Geometry. This Geometry is without doubt just an offshoot of Saccheri’s attempted proof. It can hardly be disputed, at least from internal evidence, that these authors were acquainted with Saccheri’s work either directly or indirectly. As we have said before, the new Geometry was by no means an intellectual discovery sprung like Minerva full armed from the brain of its originators. It had already in large part been previously prepared by Saccheri, and all it required was the requisite boldness to throw aside all the shackles of conservatism, or of prejudice in established order, or what you will. Two very youthful mathematicians were found with the necessary audacity and rashness, or courage, if so be, and the trick was done. The only thing necessary was simply to leap from conviction of the impossibility of demonstrating the parallel theory to assurance that it was not true, for which the ground had been thoroughly prepared before them. With this jump made, the new Geometry was ready by the simple acceptance of the propositions used by Saccheri in the third hypothesis. That they were influenced by Saccheri may be seen not only in the theorems that are alike in both, but in the fact that they accepted the viewpoint of contemporary mathematicians with regard to Saccheri’s work. This was that, while Saccheri was successful in rejecting the hypothesis of the obtuse angle, he failed with regard to the hypothesis of the acute. It was just this fact that was seized upon by the new Geometers as the foundation for their system. Both Lobatschewsky and Bolyai built their non-Euclidean Geometry on the third hypothesis of Saccheri and Lambert, viz., that of the acute angle. They never even considered the second hypothesis, that of the obtuse angle, which became later the basis of the Geometry of Riemann. In his Introduction, Lobatschewsky thus speaks of the state of the question: “We have seen that the sum of the angles of a rectilinear triangle cannot be greater than p. There still remains the assumption that it may be equal to p or less than p. Each of these two can be adopted without any contradiction appearing in the deductions made from it; and thus arise two geometries: the one, the customary, is that which until now, owing to its simplicity, agrees fully with all practical measurements; the other, the imaginary, more general and therefore more difficult in its calculations, involves the possibility of a relation between lines and angles.”10 This was merely the adoption of the then view with regard to Saccheri’s work. The only new note is the positiveness of the assertion of the possibility of the two geometries existing side by side. So despite the assertion of Halstead that the twenty-four pages of Bolyai are the most extraordinary in the history of thought, and the comment of Gauss that the work of Lobatschewsky was done in a masterly way and in a true geometrical spirit,11 we can scarcely consider these two among the great mathematical discoverers of history, or see in their work any extraordinary originality. Of its truth and value we shall shortly have occasion to speak our mind, and there it will be even less commendatory, but we hope not without good and sufficient reason: “Amicus Plato, sed magis amica veritas.” The later development of non-Euclidean Geometry has not been along the lines of Lobatschewsky and Bolyai, although this Geometry is still supposed to hold its ground theoretically. The non-Euclidean Geometry that has acquired the greatest importance at the present day is that of Riemann. Georg Friedrich Bernhard Riemann (1826-1866), a professor of mathematics in Göttingen, adopted the second hypothesis, that of the obtuse angle, and from this point of view developed a Geometry different from that of Lobatschewsky and Bolyai. His theory was brought forward in his Habilitationsschrift, or thesis for qualifying as privat-docent. His paper was read to the Philosophical Faculty of Göttingen in 1854, but was not published till 1866, after the death of the author. Its title was, Uber die Hypothesen welche der Geometrie zu Grunde liegen, or on the Hypotheses Which Form the Foundation of Geometry. Riemann considered the postulate of the straight line as fit a subject of doubt as that of parallel lines, and he substituted instead of the unlimited line of Euclid his own notion of a finite but unbounded line. He ascribed to space a constant curvature, and hence the lines of space were curved, and each line would return to the point of its origin. Substituting this notion of a curved for a straight line, he found that the hypothesis of the obtuse angle need not be rejected, since the proofs that the obtuse angle hypothesis was contradictory, were considered to repose on the notion of an infinite straight line. Then as the hypothesis of the obtuse angle became available, another non-Euclidean Geometry arose, in which the sum of the angles of the triangle is in excess of two right angles. Thus the second hypothesis of Saccheri became a new Geometry, and as Lambert and Taurinus foresaw, would find its actualization in the geometry of the sphere. Riemann was not a geometer in the accepted sense, nor did he work out the problem in a geometrical way as Bolyai and Lobatschewsky had done for their system. He was an analytical mathematician, and merely used a geometrical hypothesis as the groundwork for his Analysis. Riemannian Geometry is then nothing more than a postulate for Riemannian analytical working out of formulae, which consists in the representation of a function by means of a trigonometrical series. This part does not concern our enquiry. His fundamental hypotheses do. And we shall consider the question of the straight line, the obtuse angle, and the sum of the interior angles of the triangle in the proper place. Riemann himself did not fully work out the applications of this new Geometry, but his successors did, especially Cayley (1821-1895) and Klein (1849-1925). It was the latter who invented the names by which the two different non-Euclidean Geometries are now usually known; he called the three different geometries respectively, Hyperbolic, Elliptic, and Parabolic. The first is that of Lobatschewsky and Bolyai; the second that of Riemann; the third that of Euclid. Elliptic or Riemann’s Geometry was later divided by Klein into two separate cases, according to the character of the plane studied by the Geometry. In the case of the double elliptic or spherical plane, or plane with a character of a two sided surface, we have one; and the case of the single elliptic plane, or one that has the character of a one sided surface, gives the other. But all this is beyond the scope of the present enquiry. To these three kinds of geometry correspond three different conceptions of space and of its properties, since geometry is the science of the determinations of space. Euclidean geometry accepts the definition of straight line, of parallels, the fifth postulate, and that the sum of the interior angles of every triangle is equal to two right angles. Its space is the ordinary conceptual, three dimensional, homogeneous, isotropic space, in which extended bodies exist, and the mechanical motion of translation and rotation takes place. The second Geometry, called Imaginary Geometry by Lobatschewsky, Logarithmic-spheric by Taurinus, Hyperbolic by Klein, is built on the hypothesis of the acute angle, or the third hypothesis of Saccheri and his successors; the straight line is infinite, there are any number of non-secants through a point with regard to a set line, and the sum of the angles of a triangle is less than two right angles. This Geometry accepts the definition of parallels, but rejects the fifth postulate. The surface required in this Geometry is that shown by Lambert to be an imaginary spherical surface, which was afterwards shown analytically by Beltrami to be the geometry of an imaginary sphere, which he called sphere imaginaire. This is the locus of all points situated at the same imaginary distance from a given point. This sphere is now called the pseudo-sphere. Of course it is not a possible figure, but a so-called imaginary one, and is therefore a pure algebraic or analytic relation. If r2 (A+B+C-p) is the area of a spherical triangle with angles A, B, and C, and a radius r, the area of the imaginary spherical triangle is r2 (p-A-B-C) which is obtained by substituting instead of r2 the expression (r\/—-1)2 where the \/—-1 is an imaginary quantity. These formulae were shown to apply by Lambert.12 Lobatschewsky arbitrarily suppresses the hypothesis of the obtuse angle. Then in Prop. XIX he proves that in every right-angled triangle the sum of the three angles cannot surpass two right angles. In Prop. XX he shows that, if, in any rectilinear triangle, the sum is equal to two right angles, the same will be true of every other triangle. In Prop. XXXII he argues that a circle whose radius goes on increasing, changes at its limit into a curved line, which he calls horicycle, such that all perpendiculars erected on its chords are parallel among themselves. This curve had already been developed by Saccheri, where in the second part of his work he studies the curve that would be the locus of points at equal distances from a perpendicular, according to the third hypothesis, or that of the acute angle. The horicycle in Lobatschewskian Geometry would correspond to the straight line in the Euclidean. Bolyai and Lobatschewsky take the radius as unity. Their Geometry can be reduced to Euclidean geometry when we take the parameter of a certain value, i.e., as an infinite radius. This is also considered by both Gauss and Schweikart, who suppose that physical geometry depends on a parameter that is finite; at infinity it represents Euclidean. Infinity is however a fictitious mathematical quantity. The third, Riemannian or Elliptic Geometry, represents elliptic space. In this space every straight line is closed, that is, it returns upon itself in a curve; the sum of the angles in the triangle is taken as greater than two right angles; there are no non-secants, and two geodesics always meet. The geodesic is the shortest distance in each kind of space. In Riemann’s Geometry the geodesic corresponds to the curve of constant curvature in Euclidean space. All straight lines in Riemann’s Geometry will thus meet twice, i.e., in either direction; but there are two cases; the meeting point may be the same or different according as the Riemannian Geometry concerns the single elliptic or double elliptic plane. In Riemann’s Geometry the line is then unbounded but finite, and space likewise. The next step after Riemann in the development of non-Euclidean Geometry was by Beltrami, in his Saggio di Interpretazione della geometria non-Euclidea, where he brought Riemann’s work into connection with that of Bolyai and Lobatschewsky. He gives the analytic Euclidean interpretation of both Geometries when confined to two dimensions. Beltrami has shown that the Riemannian line is the geodesic of a Euclidean surface of constant curvature; i.e., the surface of the ordinary sphere. The plane trigonometry of Riemann is therefore the spherical trigonometry of Euclid. The Lobatschewskian line is a line of negative curvature in Euclidean geometry, and hence the plane Lobatschewskian trigonometry becomes hyperbolic Euclidean trigonometry. The further progress of non-Euclidean Geometry has altogether been in the direction of Differential and Projective Geometries, such as the differential Geometry of manifolds. It was Klein who introduced Projective Geometry into the study of non-Euclidean Geometries. There has been but one other change wrought in non-Euclidean Geometry beyond the mere further analytical development of the system, and that is the introduction of a Geometry of four dimensions. This was done by Minkowski, who made time a fourth coordinate in the determination of the space conditions of bodies and their motions. In this conception of Geometry space itself is nothing, time nothing; but events take place in a combination of both called space-time. As Minkowski himself boastfully puts it: “From henceforth time by itself and space by itself are mere shadows, they are only two aspects of a single and indivisible manner of coordinating the facts of the physical world.” Lagrange had already introduced this idea into Mechanics: “The position
of a point in space depends on three rectangular coordinates. These coordinates
in the problems of Mechanics are conceived as being functions of t. Thus
we may regard Mechanics as a Geometry of four dimensions.”13
This Riemann-Minkowski Geometry is the foundation of the Relativity theory of Einstein, who has worked out all his equations explaining the physical universe as based on this non-Euclidean theory. The Geometry of Einstein is the Geometry of Riemann, and in Einstein’s theory space is not only not Euclidean, but Riemannian, and, apart from local disturbances, that of Riemann, Klein and Newcomb. For him too the only actual thing is space-time after Minkowski’s theory. This is advanced as a unity representing the only reality connected with our notions of space and time. His formulae are based on this and on the equation of the line from Riemann, which is raised to the fourth dimension to take care of the new coordinate, time. This view of geometry is that of Einstein’s General Theory of Relativity given out in 1915. Since then (1929) he has further developed his idea in an attempt to reduce gravitation and electromagnetism to unity. To devise equations showing the relations of gravitation and electromagnetism he had to have recourse to a new Geometry. That of Riemann was not broad enough to fit the new Mathematics. His curved space with its necessarily intersecting lines did not give the requisite freedom for mathematical formulae. Einstein has now given us a space-time concept that still keeps the Riemannian curvature, which before rendered parallel lines impossible, and also gives us, along with it, parallel lines in this “four-dimensional continuum,” and this he calls parallelism-at-a-distance. Of course this has but one purpose, to enable Einstein to manipulate the necessary analytical formulae. The concepts back of it do not worry him.14 We shall wait till we have established the futility of the non-Euclidean hypothesis, before criticising all these conceptions in detail.
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