Abstract
Euclid’s fifth postulate has been accepted as a theorem since the time of ancient Greece. The efforts to prove it have been going on for nearly 2 000 years. Non-Euclidean geometry, based on its rejection, emerged in the first half of the 19th century. The author of the present article returns to the problem by addressing the metaphysical foundations of physics. The author has found the ideal instrument for analyzing infinity to be an infinitely small unit, which cannot be divided further. With the help of this instrument, the fundamental properties of the so-called space were found. It was concluded that there are no oblique or curved lines on the basic level. The apparent curved and oblique lines are stairs with negligibly fluent changing or constant steps, correspondingly. Hence, the refutation of non-Euclidean geometries and seeking a new proof of the postulate. Inter alia, it was found that the requirement to conclude the proof from Euclid’s other four axioms only diverted the attention of mathematicians from the true problem. The author proved the fifth postulate on a plane. Its application to a pair of skew lines is considered. In conclusion, the author describes the basic properties of the so-called space.
Highlights
The essentials of modern geometry were set in ancient Egypt as a totality of mechanical habits
The corpus of geometry, formed by Euclid, is the fusion of geometric constructions with axiomatic proofs, which permit the binding of complicated conclusions with the basic statements via vivid demonstrations
Elaborating on the metametric groundings of physics, and coming to conclusions incompatible with those of non-Euclidean geometry, the author was forced to return to the root of the problem, i.e., to the fifth postulate, and to comprehend the misfortunes of his predecessors, in attempting to prove the postulate
Summary
The essentials of modern geometry were set in ancient Egypt as a totality of mechanical habits. While proving the main part of the fifth postulate, the author of the present article revealed a certain similarity to the approach of Nasir al-Din Tusi (1201 – 1274) Before the proof, he prefaced three auxiliary theorems (lemmas). Elaborating on the metametric groundings of physics, and coming to conclusions incompatible with those of non-Euclidean geometry, the author was forced to return to the root of the problem, i.e., to the fifth postulate, and to comprehend the misfortunes of his predecessors, in attempting to prove the postulate. The methodological drawback of the founders of non-Euclidean geometry is striking: if they considered the fifth postulate to be unprovable, and inconsistent, contrarily they viewed their own developments as irrefutable, and as consistent. The opposite is true: the fifth postulate was not refuted, which would be the surest proof of non-Euclidean geometries.
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