The Greeks knew that numbers of the form $\sqrt{n}$ for nonsquare
integers $n$ are not rational. Much later, Lambert (1768) proved that
the values of $e^x$ and $\tan x$ are irrational for nonzero rational numbers $x$ (and conjectured that these values are transcendental, hence cannot be constructed using ruler and compass). My question is; what happened in between?
Here's what little I have found:
-
Fibonacci (Flos) showed that the real root of $x^3 + 2x^2 + 10x = 20$ is neither
rational, nor the square root of a rational, nor equal to one of the other irrational
numbers occurring in Euclid X. -
M. Stifel (Arithmetica integra, 1544) at least claimed that e.g. cube roots of noncube
integers are not rational. -
Fermat claimed to have a proof that if $a$ and $b$ are positive rational numbers such
that $a^2 + b^2 = 2(a+b)x + x^2$, then neither $x$ nor $x^2$ are rational.
Apart from occasional claims that Euler proved the irrationality of $e$ there seem to be no results in this direction between Euclid and Lambert.
Are there any irrationality proofs going beyond the square roots of integers and known before Euler and Lambert?
Edit Following up on Michael Hardy's suggestion, I haven't found anything predating Euler. On the other hand, Euler, in his Introductio in analysin infinitorum, claims that logarithms $\log_a b$ are "neither rational nor irrational" for integers $a, b > 1$. He does not prove that the logarithms are irrational (probably because he regarded it as trivial), and claims that they are not irrational (meaning it is not the square root of a nonsquare rational) since otherwise we would have $a^{\sqrt{m}} = b$, "which is impossible" (again, no proof, but this time it is not at all obvious but a very special case of Gelfond-Schneider).
Best Answer
I don't know when things like $\log_2 3$ were first proved irrational, but the proof is simpler than the proofs involving square roots of integers: If $2^n = 3^m$, then an even number equals an odd number.