Converse of Euclid's Parallel Postulate
Euclid did not postulate the converse of his fifth postulate, which is one way to distinguish Euclidean geometry from elliptic geometry. The Elements contains the proof of an equivalent statement (Book I, Proposition 27): If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another. As De Morgan pointed out, this is logically equivalent to (Book I, Proposition 16). These results do not depend upon the fifth postulate, but they do require the second postulate which is violated in elliptic geometry.
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“The American doctrinaire is the converse of the American demagogue, and, in this way, is scarcely less injurious to the public. The first deals in poetry, the last in cant. He is as much a visionary on one side, as the extreme theoretical democrat is a visionary on the other.”
—James Fenimore Cooper (1789–1851)
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