  A linear extension of a poset $P$ is a permutation of the elements of the set
that respects the partial order. Let $L(P)$ denote the number of linear
extensions. It is a #P complete problem to determine $L(P)$ exactly for an
arbitrary poset, and so randomized approximation algorithms that draw randomly
from the set of linear extensions are used. In this work, the set of linear
extensions is embedded in a larger state space with a continuous parameter ?.
The introduction of a continuous parameter allows for the use of a more
efficient method for approximating $L(P)$ called TPA. Our primary result is
that it is possible to sample from this continuous embedding in time that as
fast or faster than the best known methods for sampling uniformly from linear
extensions. For a poset containing $n$ elements, this means we can approximate
$L(P)$ to within a factor of $1 + \epsilon$ with probability at least $1 -
\delta$ using an expected number of random bits and comparisons in the poset
which is at most $O(n^3(ln n)(ln L(P))\epsilon^{-2}\ln \delta^{-1}).$
