constructed with a straightedge and a compass. But it was proved centuries after Euclid that Euclidean numbers cannot involve any operations other than addition, subtraction, multiplication, division, and the extraction of square roots. Angle trisection and doubling the cube Both trisecting the general angle and doubling the cube require taking cube roots, which are not constructible numbers by compass and straightedge. Squaring the circle is not a Euclidean number ... and therefore it is impossible to construct, by Euclidean methods a length equal to the circumference of a circle of unit diameter A proof exists to demonstrate that any Euclidean number is an algebraic number — a number that is the solution to some polynomial equation. Therefore, because was proved in 1882 to be a transcendental number and thus by definition not an algebraic number, it is not a Euclidean number. Hence the construction of a length from a unit circle is impossible, and the circle cannot be squared. Constructing an equilateral n-gon The Gauss-Wantzel theorem showed in 1837 that constructing an equilateral n-gon is impossible for most values of n. Euclid's parallel postulate The parallel postulate from Euclid's Elements is equivalent to the statement that given a straight line and a point not on that line, only one parallel to the line may be drawn through that point. Unlike the other postulates, it was seen as less self-evident. Nagel and Newman argue that this may be because the postulate concerns "infinitely remote" regions of space; in particular, parallel lines are defined as not meeting even "at infinity", in contrast to asymptotes. This perceived lack of self-evidence led to the question of whether it might be proven from the other Euclidean axioms and postulates. It was only in the nineteenth century that the impossibility of deducing the parallel postulate from the others was demonstrated in the works of Gauss, Bolyai, Lobachevsky, and Riemann. These works showed that the parallel postulate can moreover be replaced by alternatives, leading to non-Euclidean geometries. Nagel and Newman consider the question raised by the parallel postulate to be "...perhaps the most significant development in its long-range effects upon subsequent mathematical history". In particular, they consider its outcome to be "of the greatest intellectual importance," as it showed that "a proof can be given of the impossibility of proving certain propositions [in this case, the parallel postulate] within a given system [in this case, Euclid's first