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http://hem.bredband.net/b153434/Works/Einstein.htm
Divine Albert: "In a space which is free of gravitational fields we
introduce a Galilean system of reference K (x, y, z, t), and also a
system of co-ordinates K' (x', y', z', t') in uniform rotation
relatively to K. Let the origins of both systems, as well as their
axes of Z, permanently coincide. We shall show that for a space-time
measurement in the system K' the above definition of the physical
meaning of lengths and times cannot be maintained. For reasons of
symmetry it is clear that a circle around the origin in the X, Y plane
of K may at the same time be regarded as a circle in the X', Y' plane
of K'. We suppose that the circumference and diameter of this circle
have been measured with a unit measure infinitely small compared with
the radius, and that we have the quotient of the two results. If this
experiment were performed with a measuring-rod at rest relatively to
the Galilean system K, the quotient would be [pi]. With a measuring-
rod at rest relatively to K', the quotient would be greater than [pi].
This is readily understood if we envisage the whole process of
measuring from the "stationary" system K, and take into consideration
that the measuring-rod applied to the periphery undergoes a Lorentzian
contraction, while the one applied along the radius does not. Hence
Euclidean geometry does not apply to K'."

Why does Divine Albert believe that, "with a measuring-rod at rest
relatively to K', the quotient would be greater than [pi]"? Obviously
because "the measuring-rod applied to the periphery undergoes a
Lorentzian contraction". Then why does the segment of the periphery,
to which the measuring-rod is applied and which moves together with
the measuring-rod, fail to undergo a Lorentzian contraction?

In 1902, in "La Science et l'hypothese", Henri Poincare, in order to
justify non-Euclidean geometries, presented a parabole. Bidimensional
creatures live on a disc. The disc is heated under its center so that
the temperature is high at the center and decreases towards the
periphery. The creatures use rigid measuring rods in order to
determine the geometry of their world. They know nothing about the
heater and accordingly discover that the ratio of the circumference
and the diameter is greater than [pi]. The creatures conclude that
Euclidean geometry cannot be true on their disc.

Pentcho Valev
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