  Algol ($\beta$ Persei) is the prototypical semi-detached eclipsing binary and
a hierarchical triple system. From 2006 to 2010 we obtained 121 high-resolution
and high-S/N \'{e}chelle spectra of this object. Spectral disentangling yields
the individual spectra of all three stars, and greatly improved elements both
the inner and outer orbits. We find masses of $M_{\rm A} = 3.39\pm0.06$
M$_\odot$, $M_{\rm B} = 0.770\pm0.009$ M$_\odot$ and $M_{\rm C} = 1.58\pm0.09$
M$_\odot$. The disentangled spectra also give the light ratios between the
components in the $B$ and $V$ bands. Atmospheric parameters for the three stars
are determined, including detailed elemental abundances for Algol A and Algol
C. We find the following effective temperatures: $T_{\rm A} = 12\,550\pm120$ K,
$T_{\rm B} = 4900\pm300$ K and $T_{\rm C} = 7550\pm250$ K. The projected
rotational velocities are $v_{\rm A} \sin i_{\rm A} = 50.8\pm0.8$ km/s, $v_{\rm
B} \sin i_{\rm B} = 62\pm2$ km/s and $v_{\rm C} \sin i_{\rm C} = 12.4\pm0.6$
km/s. This is the first measurement of the rotational velocity for Algol B, and
confirms that it is synchronous with the orbital motion. The abundance patterns
of components A and C are identical to within the measurement errors, and are
basically solar. They can be summarised as mean metal abundances: [M/H]$_{\rm
A} = -0.03\pm0.08$ and [M/H]$_{\rm C} = 0.04\pm0.09$. A carbon deficiency is
confirmed for Algol A, with tentative indications for a slight overabundance of
nitrogen. The ratio of their abundances is (C/N)$_{\rm A} = 2.0\pm0.4$, half of
the solar value of (C/N)$_{\odot} = 4.0\pm0.7$. The new results derived in this
study, including detailed abundances and metallicities, will enable tight
constraints on theoretical evolutionary models for this complex system.
