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[5] Many other statements equivalent to the parallel postulate have been suggested, some of them appearing at first to be unrelated to parallelism, and some seeming so self-evident that they were unconsciously assumed by people who claimed to have proven the parallel postulate from Euclid's other postulates. These equivalent statements include:
However, Euclid's reasoning from assumptions to conclusions remains valid independently from the physical reality. [4] Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath): [5] Let the following be postulated:
The Elements (Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid c. 300 BC. It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions.
Euclid (/ ˈ j uː k l ɪ d /; Ancient Greek: Εὐκλείδης; fl. 300 BC) was an ancient Greek mathematician active as a geometer and logician. [2] Considered the "father of geometry", [3] he is chiefly known for the Elements treatise, which established the foundations of geometry that largely dominated the field until the early 19th century.
In this work Klügel examined 28 attempts to prove Postulate 5 (including Saccheri's), found them all deficient, and offered the opinion that Postulate 5 is unprovable and is supported solely by the judgment of our senses. The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry.
The postulates of Euclid are profitably motivated by saying that they lead to a great wealth of geometric facts. The truth of these complicated facts rests on the acceptance of the basic hypotheses. However, by throwing out Euclid's fifth postulate, one can get theories that have meaning in wider contexts (e.g., hyperbolic geometry). As such ...
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More formally, the only permissible constructions are those granted by the first three postulates of Euclid's Elements. It turns out to be the case that every point constructible using straightedge and compass may also be constructed using compass alone, or by straightedge alone if given a single circle and its center.