Today is the pi day, so I continue the brief history of $\pi$
Johann Heinrich Lambert was the first to show the irrational nature of $\pi$ in 1761 in Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques: that could be written in this way: \[\tan(x) = \cfrac{x}{1 - \cfrac{x^2}{3 - \cfrac{x^2}{5 - \cfrac{x^2}{7 - {}\ddots}}}}\] Lamberd proved that if $x$ is not null and rational, then the previous expression must be irrational. So the irrationality of $\pi$ follows from $\tan (\pi /4) = 1$. A good synthesis of Lambert's proof is on The world of $\pi$.
In 1997 Laczkovich proposed a simplification of this demonstration, while another variation was proposed in 2009 by Li Zhou, using the integral calculus. In particular the second demonstration is inspired by the proof that Charles Hermite written in two letters to Paul Gordan and Carl Borchardt in 1873. Following Harold Jeffreys in Scinetific interference (1973), a simplification of this proof, that used a reductio ab adsurdum is proposed by Mary Cartwright.
Another proof of one page about the irrationality of $\pi$ is dued by Ivan Niven in 1946.
At the other hand, the transcendence of $\pi$ is a direct consequence of the Lindemann-Weierstrass theorem:
If $\alpha_1$, $\cdots$, $\alpha_n$ are algebraic numbers that are linearly independent over rationals, then $e^{\alpha_1}$, $\cdots$, $e^{\alpha_n}$ are algebraically independent over rationals.where an algebraic number is the solution of a polynomial equation with rational coefficients.
In 1882 Lindemann, using this theorem, showed that $e$ is transcendental, and, like a consequence of the Euler's identity, also $\pi$ is transcendental.