  In $1735$ Euler \cite{1} proved that for each positive integer $k$, the
series $\zeta(2k) = \sum_{\ell=1}^{\infty} \ell^{-2k}$ converges to a rational
multiple of $\pi^{2k}$. Many demonstrations of this fact are now known, and
Euler's discovery is traditionally proven using non-elementary techniques, such
as Fourier series or the calculus of residues \cite{2}. We give an elementary
proof, similar to Cauchy's \cite{3} proof of the identity $\zeta(2) = \pi^2/6$,
only extended recursively for all values $\zeta(2k)$. Our main formula
$$\zeta(2k)=-\dfrac{(-\pi^2)^{k}}{4^{2k}-4^{k}}\left[\dfrac{4^{k}k}{(2k)!}+{\displaystyle
\sum_{\ell=1}^{k-1}}(4^{2\ell}-4^{\ell})\dfrac{4^{k-\ell}}{(2k-2\ell)!}\dfrac{\zeta(2\ell)}{(-\pi^2)^{\ell}}\right]
\phantom{spa}k = 1,2,3,\dots$$ may be derived from previously known formulae
\cite{4}. Remarkably, Apostol \cite{5} discovered a proof similar to ours, yet
arrived at a different formula, relating $\zeta(2k)$ to the Bernoulli numbers,
\`a la Euler.
