8 8 8 A P P E N D I X H are taken, the ratio of their values is fixed, but for a single vector there is no absolute value. The same happens with the displacement, and only the angle remains absolute in this way congruence is replaced by similarity. He attempts to find the Γ’s, and he finds that they can be calculated, they are more general, and they depend on four functions f. This is very in- teresting, because the electromagnetic field is also defined by four functions. In Weyl’s geometry the g’s and f’s give the theory a complete sense, and it seems natural that the ten g’s and the four f’s define the gravitational and electromagnetic fields. One must bear in mind, [however], that this solution is not natural, because is somewhat objec- tive Nature shows us that has a definite value, inasmuch as we have natural clocks in the vibrating atoms that produce spectral series with well-defined lines. In Weyl’s analysis, the f’s exist only as results of calculation. Moreover, to find the laws of pure gravitation and of electromagnetism, it was necessary to define a function H, composed of two absolutely independent parts. We thus had a dualism to avoid it, he arrived at the solution mentioned. But the has physical meaning, and, as a consequence, this solution is not satisfactory either. He sought another ingenious route, which consisted of starting with the [Γ ’s] and find- ing the metric from the parallel displacement, that is, from the affine structure of space. One must construct all the concepts based on that. In the Riemann tensor, the R’s can be expressed by the Γ ’s, which, since they are more general than the g’s, give a much more general result and so one arrives at the formula , that is, one obtains a tensor that defines the metric, and one finds some equations which contain the cosmological term. According to Eddington, this is of no use, either, because one would have to find forty conditions in order to determine them [the Γ ’s], and he did not know how to do that. If there is only a gravitational field, the problem is not difficult. In general, the Γ ’s are not symmet- ric, but are composed of a symmetric and an antisymmetric part, which squares well with the existence in nature of metric and electromagnetic fields. What is left to do is to find the necessary conditions to obtain the Γ ’s. I have lately found a natural way. I follow a variational method, and stipulate that the integral be invariant. This integral is analogous to the integral of ordinary geometry and is the sim- plest invariant possible to make its variation zero, it is necessary to vary the Γ ’s indepen- dently. I will indicate the results obtained: If there is no electromagnetic field, that is to say, if the do not exist, one obtains the gravitational equations as in the old theory and with the cosmological term. In the first approximation, if the electromagnetic field is weak, one obtains Maxwell’s equations, which is “almost a miracle.” ds2 ds2 ds2 λgik Rik = Rik vd fik