“…the mathematician uses an indirect definition of congruence, making use of the fact that the axiom of parallels together with an additional condition can replace the definition of congruence.”

— Hans Reichenbach

The Philosophy of Space and Time (1928, tr. 1957)

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Hans Reichenbach 41

American philosopher 1891–1953

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The Philosophy of Space and Time (1928, tr. 1957)

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“The attempt to avoid a direct affirmation about infinite parallel straight lines caused Euclid to phrase the parallel axiom in a rather complicated way. He realized that, so worded, this axiom lacked the self-sufficiency of the other nine axioms, and there is good reason to believe that he avoided using it until he had to. Many Greeks tried to find substitute axioms for the parallel axiom or to prove it on the basis of the other nine. …Simplicius”

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Source: Mathematical Thought from Ancient to Modern Times (1972), p. 177
Context: The attempt to avoid a direct affirmation about infinite parallel straight lines caused Euclid to phrase the parallel axiom in a rather complicated way. He realized that, so worded, this axiom lacked the self-sufficiency of the other nine axioms, and there is good reason to believe that he avoided using it until he had to. Many Greeks tried to find substitute axioms for the parallel axiom or to prove it on the basis of the other nine.... Simplicius cites others who worked on the problem and says further that people "in ancient times" objected to the use of the parallel postulate.