[160]Subject to the limitations mentioned in the previous paragraphs.[161]SeeAppendix I.[162]We do not know whether Eddington is still prepared to defend these views.[163]An exceedingly clear presentation of this point has been given by Dr. Silberstein in his remarkable books on the relativity theory.[164]Although they do not appear to be perpendicular to each other on the diagram, the time and space directionsandare perpendicular when account is taken of the peculiar geometry of space-time with which we are dealing.[165]“Space, Time and Gravitation,” p. 141.[166]This is true only in a general way, for further empirical data are necessary when we wish to consider problems involving friction and elasticity, etc. Moreover, in considering any particular problem, a knowledge of the masses and forces involved must be obtained, and this of course entails physical measurements.[167]Geodesics are also the longest distances; the essential is that they constitute extreme distances. Thus, if we take two points at random on a sphere, the great circle passing through these points is a geodesic. But we may follow this geodesic from one point to the other in two different ways: either by going along the shortest path, or else by following the line that passes through the antipodes on the other side of the sphere. In this last case, the geodesic is the longest path.[168]To confine ourselves to a three-dimensional space, if this space were spherical, stars situated at a finite distance would yet have vanishing parallaxes, so that they would appear to be at infinity. For this reason a spherical space, though finite, would manifest itself to us visually as infinite. Theoretically, also, images of stars should form at the antipodes. In an elliptical space, however, these images would coincide with the star itself.Conversely, a Lobatchewskian space, though of infinite extent, would appear to be finite, since stars at infinity would have a non-vanishing parallax, and so would appear to be situated at a finite distance.
[160]Subject to the limitations mentioned in the previous paragraphs.
[160]Subject to the limitations mentioned in the previous paragraphs.
[161]SeeAppendix I.
[161]SeeAppendix I.
[162]We do not know whether Eddington is still prepared to defend these views.
[162]We do not know whether Eddington is still prepared to defend these views.
[163]An exceedingly clear presentation of this point has been given by Dr. Silberstein in his remarkable books on the relativity theory.
[163]An exceedingly clear presentation of this point has been given by Dr. Silberstein in his remarkable books on the relativity theory.
[164]Although they do not appear to be perpendicular to each other on the diagram, the time and space directionsandare perpendicular when account is taken of the peculiar geometry of space-time with which we are dealing.
[164]Although they do not appear to be perpendicular to each other on the diagram, the time and space directionsandare perpendicular when account is taken of the peculiar geometry of space-time with which we are dealing.
[165]“Space, Time and Gravitation,” p. 141.
[165]“Space, Time and Gravitation,” p. 141.
[166]This is true only in a general way, for further empirical data are necessary when we wish to consider problems involving friction and elasticity, etc. Moreover, in considering any particular problem, a knowledge of the masses and forces involved must be obtained, and this of course entails physical measurements.
[166]This is true only in a general way, for further empirical data are necessary when we wish to consider problems involving friction and elasticity, etc. Moreover, in considering any particular problem, a knowledge of the masses and forces involved must be obtained, and this of course entails physical measurements.
[167]Geodesics are also the longest distances; the essential is that they constitute extreme distances. Thus, if we take two points at random on a sphere, the great circle passing through these points is a geodesic. But we may follow this geodesic from one point to the other in two different ways: either by going along the shortest path, or else by following the line that passes through the antipodes on the other side of the sphere. In this last case, the geodesic is the longest path.
[167]Geodesics are also the longest distances; the essential is that they constitute extreme distances. Thus, if we take two points at random on a sphere, the great circle passing through these points is a geodesic. But we may follow this geodesic from one point to the other in two different ways: either by going along the shortest path, or else by following the line that passes through the antipodes on the other side of the sphere. In this last case, the geodesic is the longest path.
[168]To confine ourselves to a three-dimensional space, if this space were spherical, stars situated at a finite distance would yet have vanishing parallaxes, so that they would appear to be at infinity. For this reason a spherical space, though finite, would manifest itself to us visually as infinite. Theoretically, also, images of stars should form at the antipodes. In an elliptical space, however, these images would coincide with the star itself.Conversely, a Lobatchewskian space, though of infinite extent, would appear to be finite, since stars at infinity would have a non-vanishing parallax, and so would appear to be situated at a finite distance.
[168]To confine ourselves to a three-dimensional space, if this space were spherical, stars situated at a finite distance would yet have vanishing parallaxes, so that they would appear to be at infinity. For this reason a spherical space, though finite, would manifest itself to us visually as infinite. Theoretically, also, images of stars should form at the antipodes. In an elliptical space, however, these images would coincide with the star itself.
Conversely, a Lobatchewskian space, though of infinite extent, would appear to be finite, since stars at infinity would have a non-vanishing parallax, and so would appear to be situated at a finite distance.