  The Johnson-Lindenstrauss Lemma allows for the projection of $n$ points in
$p-$dimensional Euclidean space onto a $k-$dimensional Euclidean space, with $k
\ge \frac{24\ln \emph{n}}{3\epsilon^2-2\epsilon^3}$, so that the pairwise
distances are preserved within a factor of $1\pm\epsilon$. Here, working
directly with the distributions of the random distances rather than resorting
to the moment generating function technique, an improvement on the lower bound
for $k$ is obtained. The additional reduction in dimension when compared to
bounds found in the literature, is at least $13\%$, and, in some cases, up to
$30\%$ additional reduction is achieved. Using the moment generating function
technique, we further provide a lower bound for $k$ using pairwise $L_2$
distances in the space of points to be projected and pairwise $L_1$ distances
in the space of the projected points. Comparison with the results obtained in
the literature shows that the bound presented here provides an additional
$36-40\%$ reduction.
