used auxiliary functions that approximated the functions for each natural number in order to prove the transcendence of in 1873. His work was built upon by Ferdinand von Lindemann in the 1880s in order to prove that eα is transcendental for nonzero algebraic numbers α. In particular this proved that π is transcendental since eπi is algebraic, and thus answered in the negative the problem of antiquity as to whether it was possible to square the circle. Karl Weierstrass developed their work yet further and eventually proved the Lindemann–Weierstrass theorem in 1885. In 1900 David Hilbert posed his famous collection of problems. The seventh of these, and one of the hardest in Hilbert's estimation, asked about the transcendence of numbers of the form ab where a and b are algebraic, a is not zero or one, and b is irrational. In the 1930s Alexander Gelfond and Theodor Schneider proved that all such numbers were indeed transcendental using a non-explicit auxiliary function whose existence was granted by Siegel's lemma. This result, the Gelfond–Schneider theorem, proved the transcendence of numbers such as eπ and the Gelfond–Schneider constant. The next big result in this field occurred in the 1960s, when Alan Baker made progress on a problem posed by Gelfond on linear forms in logarithms. Gelfond himself had managed to find a non-trivial lower bound for the quantity where all four unknowns are algebraic, the αs being neither zero nor one and the βs being irrational. Finding similar lower bounds for the sum of three or more logarithms had eluded Gelfond, though. The proof of Baker's theorem contained such bounds, solving Gauss' class number problem for class number one in the process. This work won Baker the Fields medal for its uses in solving Diophantine equations. From a purely transcendental number theoretic viewpoint, Baker had proved that if α1, ..., αn are algebraic numbers, none of them zero or one, and β1, ..., βn are algebraic numbers such that 1, β1, ..., βn are linearly independent over the rational numbers, then the number is transcendental. Other techniques: Cantor and Zilber In the 1870s, Georg Cantor started to develop set theory and, in 1874, published a paper proving that the algebraic numbers could be put in one-to-one correspondence with the set of natural numbers, and thus that the set of transcendental numbers must be uncountable. Later, in 1891, Cantor used his more familiar diagonal