By Michal A.D., Botsford J.L.

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**Extra resources for An Extension of the New Einstein Geometry**

**Sample text**

The Ptolemaic conception of the order and machinery of the planets, the most powerful application of Greek geometry to the physical world, thus corroborated the result of direct measurement and established the dimensions of the cosmos for over 1,000 years. As the ancient philosophers said, there is no truth in astronomy. The post-classical period Passage Through Islam Two centuries after they broke out of their desert around Mecca, the followers of Muhammad occupied the lands from Persia to Spain and settled down to master the arts and sciences of the peoples they had conquered.

Within this interpretation, Euclid’s fifth postulate was an empirical finding, while nonEuclidean geometries did not apply to the real world. Bolyai apparently could not free himself from the persuasion that Euclidean geometry represented reality. Lobachevsky observed that, if there were a star so distant that its parallax was not observable from the Earth’s orbit, his geometry would be indistinguishable from Euclid’s at the point where the parallax vanished. By his calculation, based on stellar parallaxes then just detected, his geometry could be physically meaningful only in gargantuan triangles spanning interstellar space.

Pythagorean Theorem For a triangle △ABC the Pythagorean theorem has two parts: (1) if ∠ACB is a right angle, then a2 + b2 = c2; (2) if a2 + b2 = c2, then ∠ACB is a right angle. For an arbitrary triangle, the Pythagorean theorem is generalized to the law of cosines: a2 + b2 = c2 − 2ab cos (∠ACB). When ∠ACB is 90 degrees, this reduces to the Pythagorean theorem because cos (90°) = 0. Since Euclid, a host of professional and amateur mathematicians have found more than 300 distinct proofs of the Pythagorean theorem.