E. The cosmological principle
The notion that the Earth is not at the center of the Universe is generally referred to as the "Copernican Principle", though it traces its origins back to Aristarchus who thought that the Sun and the stars were in fact fixed, with the stars being at great distances.
The modern notion that the Universe on the very largest scales should be homogeneous and isotropic appears to have originated with Einstein (1917). At that time there could have been no observational basis for this assumption. However, homogeneity is a consequence of the notion that we are not in a special place in the Universe and the assumptions of homogeneity and isotropy provide for easy solutions of the Einstein field equations. The first cosmological models of Einstein and of de Sitter were based on this principle. Robertson and Walker derived their famous solution of the Einstein equations using only that principle.
It was frequently stated in the years that followed that the Universe in the large looked homogeneous and isotropic. The first systematic study was Hubble (1926) who used a sample of 400 galaxies with magnitudes, the sample was thought to be complete to magnitude 12.5. He found his counts fitted the relationship
and concluded, importantly, that "The agreement between observed and computed log N over a range of more than 8 mag. is consistent with the double assumption of uniform luminosity and uniform distribution or, more generally, indicates that the density function is independent of the distance." He goes on to look at systematics in the residuals in this plot and concludes that they may be due to "... clustering of nebulae in the vicinity of the galactic system. The cluster in Virgo alone accounts for an appreciable part."
Hubble only had data to magnitude 12. Anyone looking at the considerably fainter Shane and Wirtanen's isoplethic maps of galaxy counts based on the Lick Sky Survey (Shane and Wirtanen (1967)), or the more recent Center for Astrophysics (CfA-II) slices data (Geller and Huchra, 1989) might be forgiven for questioning the homogeneity conjecture!
The first demonstration of homogeneity in the galaxy distribution was probably the observation by Peebles that the (projected) two-point correlation function estimated from diverse catalogs probing the galaxy distribution to different depths followed a scaling law that was consistent with homogeneity. The advent of automated plate-measuring machines provided deeper and more reliable samples with which to confirm the uniform distribution number-magnitude relationship. However, at the faintest magnitude levels, these counts show significant systematic deviations from what is expected from a uniform distribution: these deviations are due to the effects of galaxy evolution at early times and their interpretation depends on models for the evolution of stellar populations in galaxies. Recently, very deep studies (Metcalfe et al. 2001) show convincingly "... that space density of galaxies may not have changed much between z = 0 and z = 3".
The first incontrovertible proof of cosmic isotropy came only as recently as early 1990s from the COBE satellite all-sky map of the cosmic microwave background radiation (Smoot et al., 1992). This map is isotropic to a high degree, with relative intensity fluctuations only at the level of 10-5. With this observation, and with the reasonable hypothesis that the Universe looks the same to all observers (the Copernican Principle) we can deduce that the Universe must be locally Friedman-Robertson Walker, ie: homogeneous as well as isotropic (Ehlers et al., 1968).