Non-Euclidean geometry

“ It will perhaps be evident now why in a sense the discovery of the non-Euclidean geometries was a stepping-stone to the consideration of hyperspace; though we should bear in mind that the two conceptions are entirely distinct, neither one being dependent upon the other. The logical conception of non-Euclidean geometry is far more difficult than the abstract notion of the fourth dimension. ”

“ It is impossible to show more clearly that the geometry we attribute to space is nothing but an expression of the properties of our bodies and measuring rods when displaced, since here we have a plane which is Euclidean or Riemannian in its geometry according to the behaviour of the bodies that glide over its surface. And so we see that two alternative methods of interpreting non-Euclidean geometry are open to us. Either we may regard our rods as Euclidean and applied to a curved surface, or else we may attribute non-Euclideanism to the behaviour of our rods on a flat surface. ”

“ For the present, let us note that since our judgment of straightness is contingent on the disclosures of experience, even the geometry of the space in which we actually live cannot be decided upon a priori. To a first approximation, to be sure, this geometry appears to be Euclidean; but we cannot prophesy what it may turn out to be when nature is studied with ever-increasing refinement. It was with this idea in view that Gauss, who had mastered in secret the implications of non-Euclidean geometry, undertook triangulations with light rays over a century ago. ”

“ I do not think that the six extra dimensions have any stalwart defenders; but we often meet with attempts to reimpose Euclidean geometry on the world in another way. The proposal, which is made quite unblushingly, is that since our measured lengths do not obey Euclidean geometry we must apply corrections to them—cook them—till they do. A closely related view often advocated is that space is neither Euclidean nor non-Euclidean; it is all a matter of convention and we are free to adopt any geometry we choose. ”

“ The non-Euclidean geometry embraces all the results obtained as a consequence of efforts made at finding a satisfactory proof of the parallel-postulate and is, therefore, based upon a conception of space which is at variance with that held by Euclid. According to the Ionian school space is an infinite continuum possessing uniformity throughout its entire extent. The non-Euclideans maintain that space is not an infinite extension; but a finite though unbounded manifold capable of being generated by the movement of a point, line or plane in a direction without itself. ”

“ People unaccustomed to non-Euclidean geometry may feel that, even if such a thing be logically possible, the world simply cannot be so odd as all that. We all have a tendency to think that the world must conform to our prejudices. The opposite view involves some effort of thought, and most people would die sooner than think—in fact, they do so. But the fact that a spherical universe seems odd to people who have been brought up on Euclidean prejudices is no evidence that it is impossible. ”

“ We can no more settle the laws of space by intuition than we can settle the laws of heredity. If intuition is ruled out, the appeal must be to experiment—genuine open-minded experiment unfettered by any preconception as to what the verdict ought to be. We must not afterwards go back on the experiments because they make out space to be very slightly non-Euclidean. It is quite true that a way out could be found. By inventing extra dimensions we can make the non-Euclidean geometry of the world depend on a Euclidean geometry of ten dimensions ”

“ You keep some notion of a space which is superior to measurement, and are ready to throw over the measures rather than let this space be distorted. Even if there were reason for believing in such a space, what possible reason could there be for assuming it to be Euclidean? Your sole reason for believing space to be Euclidean is that hitherto your measures have made it appear so; if now measures of certain parts of space prefer non-Euclidean geometry, all reason for assuming Euclidean space disappears. Mathematically and conceptually Euclidean and non-Euclidean space are on the same footing ”

“ Owing to the paths devised for the light rays and to the sameness in the reduction of the sizes of all objects as the centre was left behind, the expanding and contracting bodies of this universe would present all the characteristics of rigidity. On conducting measurements with their rigid rods the hypothetical beings would obtain non-Euclidean results, their entire world would appear to them as non-Euclidean, and non-Euclidean geometry would be as inevitable to them as Euclidean geometry is to the average layman. ”

“ The first to publish a non-Euclidean geometry was Nicholas Lobatchewsky, professor of mathematics in the new university of Kazañ.8 In the place of the disputed postulate he puts the following: “All straight lines which, in Lobatchewsky. a plane, radiate from a given point, can, with respect to any other straight line in the same plane, be divided into two classes, the intersecting and the non-intersecting. The boundary line of the one and the other class is called parallel to the given line.” ”

“ Now since axiomatic geometry by itself contains no assertions as to the reality which can be experienced, but can do so only in combination with physical laws, it should be possible and reasonable—whatever may be the nature of reality—to retain Euclidean geometry. For if contradictions between theory and experience manifest themselves, we should rather decide to change physical laws than to change axiomatic Euclidean geometry. ”

“ In projective geometry any two straight lines in a plane intersect, and the straight lines are closed series which return into themselves, like the circumference of a circle. In descriptive geometry two straight lines in a plane do not necessarily intersect, and a straight line is an open series without beginning or end. Ordinary Euclidean geometry is a descriptive geometry; it becomes a projective geometry when the so-called “points at infinity” are added. ”

“ The advantages of what is sometimes called intrinsic geometry are united to those resulting from the use of the regular analysis. Besides, this analysis is by no means abandoned; the complications of calculation which it almost always carries with it, in its applications to the study of surfaces and rectilinear coordinates, usually disappear if one employs the notion on the invariants and the covariants of quadratic powers of differentials which we owe to the researches of Lipschitz and Christoffel, inspired by Riemann's studies on the non-Euclidean geometry. ”

“ One geometry cannot be more true than another; it can only be more convenient. Now, Euclidean geometry is, and will remain, the most convenient: 1st, because it is the simplest, and it is not so only because of our mental habits or because of the kind of direct intuition that we have of Euclidean space; it is the simplest in itself, just as a polynomial of the first degree is simpler than a polynomial of the second degree; 2nd, because it sufficiently agrees with the properties of natural solids, those bodies which we can compare and measure by means of our senses. ”

“ There is no help for it, if the longest track can be a spiral like that known to be described by the earth. Non-Euclidean geometry is necessary. In Euclidean geometry the shortest track is always a straight line; and the slight modification of Euclidean geometry described in Chapter iii is found to give a straight line as the longest track. ”

“ But if the doctrine means that, assuming some sort of objective reality for the material universe, beings can be imagined, to whom either all congruence-groups are equally important, or some other congruence-group is specially important, the doctrine appears to be an immediate deduction from the mathematical facts. Assuming a definite congruence-group, the investigation of surfaces (or three-dimensional loci in space of four dimensions) with geodesic geometries of the form of metrical geometries of other types of congruence-groups forms an important chapter of non-Euclidean geometry. ”

“ Non-Euclidean geometry, however, showed other possibilities. The surface of a sphere has no boundary, yet it is not infinite. In [Pg 165] traveling round the earth, we never reach “the edge of the world,” and yet the earth is not infinite. The surface of the earth is contained in three-dimensional space, but there is no reason in logic why three-dimensional space should not be constructed on an analogous plan. What we imagine to be straight lines going on for ever will then be like great circles on a sphere: they will ultimately return to their starting point. ”

“ The existence of a natural unit of length is a peculiarity common both to hyperbolic and elliptic geometries, and differentiates them from Euclidean geometry. It is the reason for the failure of the theory of similarity in them. If γ is very large, that is, if the natural unit is very large compared to the arbitrary unit, and if the lengths involved in the figures considered are not large compared to the arbitrary unit, then both the elliptic and hyperbolic geometries approximate to the Euclidean. ”

“ The great geometer Legendre and other mathematicians attempted to give a proof of this so-called axiom, but without success. At last, to make a long story short, some mathematicians began to believe that this proposition was not only not self-evident, but was not capable of proof, and moreover that an equally consistent geometry could be built up on the supposition that it is not always true. Thus, out of various endeavors to prove Euclid's "parallel" axiom, arose non-Euclidean geometry, the beginning of which is sometimes attributed to Gauss ”

“ There is many a rational logos, and the mathematician has high delight in the contemplation of inconsistent systems of consistent relationships. There are, for example, a Euclidean geometry and more than one species of non-Euclidean. As theories of a given space, these are not compatible. If our universe be, as Plato thought, and nature-science takes for granted, a space-conditioned, geometrised affair, one of these geometries may be, none of them may be, not all of them can be, valid in it. ”

“ We showed that, although concepts might be originated by sensations, they were not, nor were affected by, external facts. The relation between mathematical truths and external facts is one of more or less resemblance, not of identity. Nor can the resemblance be ever proved to be perfect.
The Euclidean geometry has as great facility in accommodating itself to all known facts as the transcendental, and greater simplicity. It is therefore of greater practical utility. The mathematical truth of each is not affected by experience. ”

“ If we deny the relation between the practically-rigid body and geometry, we shall indeed not easily free ourselves from the convention that Euclidean geometry is to be retained as the simplest. Why is the equivalence of the practically-rigid body and the body of geometry — which suggests itself so readily — denied by Poincare 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. ”

“ We must never forget that we are dealing here with a problem of physics, not with one of pure mathematics. For instance, in Euclidean geometry the mathematician does not have to appeal to measurements with rigid rods in order to determine the value of , since in virtue of our basic Euclidean postulates and the rules of logical reasoning, can be determined as accurately as we wish by purely mathematical means. But in the case of the geometry of physical space we are completely at sea. ”

“ A four-dimensional continuum described by the "co-ordinates" , , , , was called "world" by Minkowski, who also termed a point-event a "world-point." From a "happening" in three-dimensional space, physics becomes, as it were, an "existence" in the four-dimensional "world."
This four-dimensional "world" bears a close similarity to the three-dimensional "space" of (Euclidean) analytical geometry. If we introduce into the latter a new Cartesian co-ordinate system (', ', ') ”

“ But as the construction progresses it becomes more and more patent that the disposition of the discs in the manner indicated, without interruption, is not possible, as it should be possible by Euclidean geometry of the the plane surface. In this way creatures which cannot leave the spherical surface, and cannot even peep out from 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. ”

“ It is evident that in the Euclidean geometry only such, and all such, quantities have an objective significance, independent of the particular choice of the Cartesian co-ordinates, as can be expressed by an invariant with respect to linear orthogonal transformations. This is the reason that the theory of invariants, which has to do with the laws that govern the form of invariants, is so important for analytical geometry. ”

“ Of course if, as in the hypothetical world discussed previously, material bodies behaved like non-Euclidean solids and if light rays followed appropriate courses, we should have to abandon Euclidean geometry for reasons of practical convenience. But since, in the world we live in, our habitual solids behave to a high order of approximation as do Euclidean solids, our preference for Euclidean geometry seems perfectly legitimate even from the standpoint of physics. ”