Page "Diophantine approximation" Paragraph 3

It appears that these bounds depends on the nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number is larger that the lower bound for algebraic numbers, which is itself larger than the lower bound for all real numbers.

Thus a real number that may be better approximated than the bound for algebraic numbers is certainly a transcendental number.

This allowed Liouville, in 1844 to produce the first explicit transcendental number.

Later, the proofs that