Whereas Euclid's approach to geometry was additive (he started with basic definitions and axioms and proceeded to build a sequence of results depending on previous ones), Klein's approach was subtractive. Instead, we will approach the subject as the German mathematician Felix Klein (\(1849\)-\(1925\)) did. This text does not develop geometry as Euclid, Lobachevsky, and Bolyai did. Non-Euclidean geometry was thus placed on solid ground. By changing Euclid's parallel postulate, was a system created that led to contradictory theorems? In \(1868\), the Italian mathematician Enrico Beltrami (\(1835\)-\(1900\)) showed that the new non-Euclidean geometry could be constructed within the Euclidean plane so that, as long as Euclidean geometry was consistent, non-Euclidean geometry would be consistent as well. One of the first challenges of non-Euclidean geometry was to determine its logical consistency. It is amazing that the Parallel Postulate, being equivalent to such intuitive statements as 1 and 8 above, is also equivalent to the Pythagorean Theorem. Ivan discourages his younger brother from thinking about whether God exists, arguing that if one cannot fathom non-Euclidean geometry, then one has no hope of understanding questions about God. By all accounts, the Pythagorean Theorem is far from obvious. Early in the novel two of the brothers, Ivan and Alyosha, get reacquainted at a tavern. Fyodor Dostoevsky thought non-Euclidean geometry was interesting enough to include in The Brothers Karamazov, first published in \(1880\). The arrival of non-Euclidean geometry soon caused a stir in circles outside the mathematics community.
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