The {\em abeliant} is a polynomial rule for producing an $n$
by $n$ matrix with entries in a given ring from an
$n$ by $n$ by $n+2$ array of elements of that ring.
The theory of abeliants, first introduced in an
earlier paper of the author, is redeveloped here in a
simpler way. Then this theory is exploited to give an explicit
elementary construction of the Jacobian of a nonsingular
projective algebraic curve defined over an
algebraically closed field. The standard of usefulness and
aptness we strive toward is that set by Mumford's elementary
construction of the Jacobian of a hyperelliptic curve.
This paper has appeared as Advances in Math 172 (2002) 169-205.