Who first proved that there are at least n^(1-ε) primes up to n? It's well-known that Hadamard and de la Vallée-Poussin independently proved the Prime Number Theorem in 1896: that $\pi(n)=n/\log n+o(n/\log n)$.  I'm curious as to a weaker result: that for any $\varepsilon>0$, $\pi(n)\gg n^{1-\varepsilon}$.
Chebyshev famously proved that if $\lim \pi(n)\log n$ exists it must be equal to 1, but I seem to remember that he also proved bounds on that value, pushing the date back to 1850 or so in that case.  But were there earlier results in this direction?
 A: Didn't Euler prove that $\sum 1/p$ diverges? This proves that $\pi(N)$ is infinitely often larger than $N^{1-\epsilon}$: If there were only $N^{1-\epsilon}$ primes less than $N$, then there are at most $2^{k (1-\epsilon)}$ primes between $2^{k-1}$ and $2^k$. So we can bound $\sum 1/p$ above by $\sum 2^{k(1-\epsilon)}/2^{k-1} = 2 \sum 2^{-k \epsilon}$, which converges.
UPDATE: I should clarify that I see no way to get from here to the stronger statement that $\pi(N)$ is greater than $N^{1 - \epsilon}$ for all sufficiently large $N$.
A: In the 1850's, Chebyshev gave the following explicit bound, for sufficiently large n.
$$0.92129 \, \frac{n}{\log n} < \pi(n) < 1.0556 \, \frac{n}{\log n}.$$ 
This is mentioned in "An introduction to the theory of the Riemann zeta function" by S. J. Patterson. There is an exercise in the first chapter that gives the lower bound following Chebyshev's method (that is, using Stirling's formula and the series $\log \ n! = \Sigma_{p^k\leq n} \lfloor n/p^k \rfloor\log p$). S. J. Patterson cites Chebyshev as the first to obtain significant results toward the prime number theorem.
A: From MathWorld's article on the PNT:

In 1792, when only 15 years old, Gauss
  proposed that $\pi(n) \sim n/\log n$.
Gauss later refined his estimate to
  $\pi(n) \sim \mbox{Li}(n)$ where 
$\mbox{Li}(n) := \int_2^n \frac{1}{\log x} \, dx$ 
is the logarithmic integral. Gauss did
  not publish this result, which he
  first mentioned in an 1849 letter to
  Encke. It was subsequently
  posthumously published in 1863.

