Paul Dirac attended the first 10 physics meetings organised in Lindau and gave talks at all except one. In 1979 he choose to talk about a subject based on a more than 40-year old love of his, the so-called Large Numbers Hypothesis from 1937. This hypothesis emanated from the fact that the strength of the electric force today is about 40 orders of magnitude larger than the gravitational force and that this is of the same order of magnitude as the age of the universe (in atomic time units). In his lecture Dirac assumes that the two numbers always have been proportional to each other. From this he draws a number of interesting conclusions. One of them is that the gravitational constant varies with the age of the universe and is decreasing. Another conclusion is that the only viable model of the universe is based on the one that Albert Einstein and Willem de Sitter published in 1932. Dirac’s assumptions lead to effects that should be observable. One of them is a difference between time as measured by an atomic clock and time as measured by the motion of the earth. Another effect is a very slow spiralling of the planetary orbits towards the sun. Due to a revival of interest in cosmology in the 1950’s, observational techniques for high-precision measurements of astrophysical effects had been developed. Irwin Shapiro had bounced radar signals on the planets and measured different parameters and the Apollo collaboration had bounced laser pulses from the moon. At the end of his lecture, Dirac discusses the status of the observations and concludes that they cannot yet confirm or rule out his theory.