Euclidean space

“ Obviously we must start with a four-dimensional Euclidean space in which a three-dimensional spherical surface is embedded. But as a four-dimensional Euclidean space transcends our immediate experience and is in the nature of a mathematical fiction, we should be led to the erroneous conclusion that three-dimensional non-Euclidean geometry must likewise be a purely conceptual construction having nothing in common with measurements which might be conducted with material rods. ”

“ 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. ”

“ As for Euclidean congruence and Euclidean rigidity, it is by no means more representative of real rigidity than are the non-Euclidean varieties. There is therefore no reason to appeal to a distorting effect of empty space in order to account for a possible non-Euclidean behaviour of our material solids when displaced from point to point. Non-Euclideanism may or may not exist in real space, but this is a point for physical measurement and not for philosophy or mathematics to decide. ”

“ If we proceed beyond the domain of two-dimensional geometry we merge the ideas of non-Euclidean and hyper-space. The ordinary triply-extended space of our experience is purely Euclidean; and if we approach the conception of curvature in such a space it must be curvature in a fourth dimension, and here the mind refuses to follow, although by pure reasoning we can show what must take place in such a space. ”

“ 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. ”

“ Beings whose minds were made as ours, and with senses like ours, but without any preliminary education, might receive from a suitably-chosen external world impressions which would lead them to construct a geometry other than that of Euclid, and to localise the phenomena of this external world in a non-Euclidean space, or even in space of four dimensions. As for us, whose education has been made by our actual world, if we were suddenly transported into this new world, we should have no difficulty in referring phenomena to our Euclidean space. ”

“ At a great distance from gravitating matter, we can so choose our co-ordinates as to obtain a Euclidean space; this is really only another way of saying that the special theory of relativity applies. In the neighborhood of matter, although we can make our space Euclidean in any small region, we cannot do so throughout any region within [Pg 127] which gravitation varies sensibly—at least, if we do, we shall have to abandon the view that bodies move in geodesics. In the neighborhood of a piece of matter, there is, as it were, a hill in space-time ”

“ The Euclidean space may, like our kind of reason, be simply an idiosyncrasy of certain kinds of animals—other kinds might find necessary a space of four dimensions and have a different type of logic from the human. ”

“ This synthesis or association of abstracts has nothing necessary about it; its elements may be combined in several ways; hence the possibility of different concepts of space (Euclidean, non-Euclidean) . Space conceived as infinity reduces itself to the power that the human mind [159] has of forming sequences, and it forms them thanks to abstraction, which admits of its seizing the law of their formation.
Intuition is the common basis of all concepts of space. Euclidean space rests directly upon this, and upon definitions. ”

“ I cannot imagine myself perceiving non-Euclidean space!
Math. Look at the reflection of the room in a polished doorknob, and imagine yourself one of the actors in what you see going on there. ”

“ 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. ”

“ 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. ”

“ There will not be in the universe anything straighter than these great circles; the Euclidean straight line may remain as a beautiful dream, but not as a possibility in the actual world. In particular, light rays in empty space will travel in what are really great circles. If we could make measurements with sufficient accuracy, we should be able to infer this state of affairs even from a small part of space, because the sum of the angles of a triangle would always be greater than two right angles, and the excess would be proportional to the size of the triangle. ”

“ AS non-Euclidean geometry has not become popular enough to find a place in the ordinary college curriculum, and as its discovery preceded any serious consideration of space of more dimensions than three, it seems to me that, before taking up hyperspace proper, it would be well at least to mention the non-Euclidean geometries. ”

“ 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. ”

“ We have apparently proved that up to a certain scale of magnitude optical space is Euclidean with respect to measures of angle, but this may not necessarily involve that space is also Euclidean with respect to measures of length, so that space need not be completely Euclidean. ”

“ The same questions which came up apropos of the truths of Euclidean geometry, come up anew apropos of the theorems of analysis situs. Are they obtainable by deductive reasoning? Are they disguised conventions? Are they experimental verities? Are they the characteristics of a form imposed either upon our sensibility or upon our understanding?
I wish simply to observe that the last two solutions exclude each other. We can not admit at the same time that it is impossible to imagine space of four dimensions and that experience proves to us that space has three dimensions. ”

“ Everything in physical science, from the law of gravitation to the building of bridges, from the spectroscope to the art of navigation, would be profoundly modified by any considerable inaccuracy in the hypothesis that our actual space is Euclidean. The observed truth of physical science, therefore, constitutes overwhelming empirical evidence that this hypothesis is very approximately correct, even if not rigidly true.—Russell, Bertrand. ”

“ If his right to appropriate the term space in this way is challenged, he would urge that this is the sense in which the term has always been used in physics hitherto; it is only recently that conservative physicists, frightened by the revolutionary consequences of modern experiments, have begun to play with the idea of a pre-existing space whose properties cannot be ascertained by experiment—a metaphysical space, to which they arbitrarily assign Euclidean properties, although it is obvious that its geometry can never be ascertained by experiment. ”

“ 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. ”

“ In view of all these occurrences, difficult to account for if we believe in the amorphous nature of space, unless we appeal to some miraculous pre-established harmony, it appears as though space must be credited with a definite structure or metrics which, in the [Pg 57] light of experiment, turns out to be Euclidean, at least to a first approximation. Expressed in a different way, real space appears to be permeated by an invisible field, the Metrical Field, endowing it with a metrics or structure. ”

“ 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 ”

“ 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. ”

“ We choose to impose the additional condition that the resulting metric space be a Euclidean geometry with a rectangular coordinate system. [Pg 67]
Even this always does not yield uniqueness, but we will show the additional restriction that will guarantee uniqueness after the necessary language is developed. Since all metrizations of a given metrizable topology are isomorphic, in the quotient class the orthogonal Euclidean geometry serves the purpose of being a convenient representative of the unique element resulting from a given metrizable topology. ”

“ These three quantities are sometimes called potentials.
We now come to a point of far-reaching importance. The formula for teaches us not only the character of the mesh-system, but the nature of our two-dimensional space, which is independent of any mesh-system. If satisfies any one of the first three formulae, then the space is like a flat surface; if it satisfies the last formula, then the space is a surface curved like a sphere. Try how you will, you cannot draw a mesh-system on a flat (Euclidean) surface which agrees with the fourth formula. ”

“ Gradually the student will become conscious that the portion of the real space which lies beyond the visible stellar universe is not cognizable through the senses, that we know nothing of its properties and consequently have no basis for judgments concerning it. Mathematical space, on the other hand, may be subjected to conditions, for instance, we may condition its properties at infinity, and these conditions constitute the axioms, say the Euclidean axioms. ”

“ The chief value, then, of non-Euclidean geometry seems to abide in the fact that it clarifies our understanding as to the complex processes by which it is possible to organize and systematize our spatial experiences for assimilation and use in other branches of knowledge. ”

“ Another frame which we impose on the world is space. Whence come the first principles of geometry? Are they imposed on us by logic? Lobachevski has proved not, by creating non-Euclidean geometry. Is space revealed to us by our senses? Still no, for the space our senses could show us differs absolutely from that of the geometer. Is experience the source of geometry? A deeper discussion will show us it is not. We therefore conclude that the first principles of geometry are only conventions; but these conventions are not arbitrary and if transported into another world ”