Leonhard Euler knew that the infinite product:

$$ \prod_{p \textrm{ prime}} \left(1 - \frac{1}{p} \right)^{-1} = \sum_{n=1}^{\infty} \frac{1}{n} $$

is divergent (and used this to prove the infinitude of primes), so would have also known that the product:

$$ \prod_{p \textrm{ prime}} \left(1 - \frac{1}{p} \right) $$

tends to zero. In other words:

$$ \lim_{n \rightarrow \infty} \prod_{p < n\textrm{ prime}} \left(1 - \frac{1}{p} \right) = 0 $$

Observe that the product in the last expression is the density of integers which are coprime to the set $\{ p < n \textrm{ such that } p \textrm{ is prime} \}$; this trivially implies that primes have zero upper density.