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A brief history of the connection between observation and theory of the ``particles'' in the universe was given in Sect. 2.3. The final comment here concerns the observational tests for the theory discussed in Sects. 3 and 4.

McCrea (1935) summarized:

``This paper has sought to make clear what particular set of assumptions are being tested when comparison is made with any particular set of observational results. The broad conclusions to which we are led are these:
If we compare a relation like that connecting apparent size, apparent brightness, and redshift we are testing merely the correctness of our interpretation of the observed quantities, and not any particular theory of them.
If we make all possible observations on the ``distances'', red-shifts, and numbers of extra-galactic nebulae we are testing the possibility of representing them as fundamental particles in a universe of the type (1) [with Friedmann-Lemaître-Robertson-Walker metric], and the correctness of the derivation of `world-pictures' in such a model.
In order however to choose between such models and classical ones it will in general be necessary to test....terms in k in expressions involving the numbers of nebulae . . .''.

Thus the connection is made to the most recent suggestion concerning k / R2 measurements (Sect. 3.2.1).

One diagram of each type of relation mentioned by McCrea will be shown in the following.