Editing Euclidean geometry (section)

===18th century===
Geometers of the 18th century struggled to define the boundaries of the Euclidean system. Many tried in vain to prove the fifth postulate from the first four. By 1763, at least 28 different proofs had been published, but all were found incorrect.<ref>Hofstadter 1979, p.&nbsp;91.</ref>

Leading up to this period, geometers also tried to determine what constructions could be accomplished in Euclidean geometry. For example, the problem of [[trisecting an angle]] with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools. However, centuries of efforts failed to find a solution to this problem, until [[Pierre Wantzel]] published a proof in 1837 that such a construction was impossible. Other constructions that were proved impossible include [[doubling the cube]] and [[squaring the circle]]. In the case of doubling the cube, the impossibility of the construction originates from the fact that the compass and straightedge method involve equations whose order is an integral power of two,<ref>Theorem 120, Elements of Abstract Algebra, Allan Clark, Dover, {{ISBN|0-486-64725-0}}.</ref> while doubling a cube requires the solution of a third-order equation.

[[Leonhard Euler|Euler]] discussed a generalization of Euclidean geometry called [[affine geometry]], which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become meaningless) while retaining the notions of parallelism as an [[equivalence relation]] between lines, and equality of length of parallel line segments (so line segments continue to have a midpoint).