D O C . 3 8 7 T H E N E W F I E L D T H E O R Y 5 7 5 Euclidean which is richer in formal relationships that the former, but poorer for the latter. Cons- dier a two-dimensional Riemannian space in the form of the surface of a hen’s egg. Since this surface is embedded in our (accurately enough) Euclidean space, it possesses a Riemannian metric. In fact, it has a perfectly definite meaning to speak of the distance of two neighboring points P, Q on the surface. Similarly it has, of course, a meaning to say of two such pairs of points (PQ) (P1Q1), at separate parts of the surface of the egg, that the distance PQ is equal to the distance P1Q1. On the other hand, it is impossible to now compare the direction PQ with the direction P1Q1. In particular it is meaningless to demand that P1Q1 shall be parallel to PQ. In the corresponding Euclidean geometry of the plane, directions can be compared and the re- lationship of parallelism can exist between lines in regions of the plane at any distance from one another (distant parallelism). To this extent the Euclidean continuum is richer in relationships than the Riemannian. The new unitary field theory is based on the following mathematical discovery: There are con- tinua with a Riemannian metric and distant parallelism which nevertheless are not Euclidean. It is easy to show, for instance, in the case of three-dimensional space, how much a continuum differs from a Euclidean. First of all, in such a continuum there are lines whose elements are parallel to one another. We shall call those “straight lines”. It also has a definite meaning to speak of two parallel straight lines as in the Euclidean case. Now [image] choose two such parallels E1L1 and E2L2 and mark on each a point P1,P2. On E1L1 choose in addition a point Q1. If we now draw through Q1 a straight line Q1-R parallel to the straight line P1,P2, then in Euclidean geometry this will cut the straight line E2L2 in the geometry now used the line Q1-R and the Line E2L2 do not in general cut one another. To this extent the geometry now used is not only a specialization of the Riemannian but also a generalization of the Euclid- ean geometry. My opinion is that our space-time continuum has a structure of the kind here out- lines. The mathematical problem whose solution, in my view, leads to the correct field laws is to be formulated thus: Which are the simplest and most natural conditions to which a continuum of this kind can be subjected? The answer to this question which I have attempted to give in a new paper yields unitary field laws for gravitation and electro-magnetism. (Concluded.) The first article appeared in the Times of yesterday.REPT. [4 034]. First published in The New York Times, 3 February 1929, then in The Times (London) (4 and 5 February 1929). PD. Published in The Observatory 52 (1930): 82–87, 114–118. For annotation, see Doc. 395. [1] Dated by the dates of the publication of Einstein 1929n (Doc. 365) and its first English publica- tion in The Times (London). [7] [5] [6]
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