Did Jacobi prove the product formula $\Delta(\tau) = (2\pi)^{12}q\prod_{n \geq 1} (1 - q^n)^{24}$ for the modular discriminant function, and if so where?

I have tried without success to track down references (other than looking through Jacobi's collected works), but when I search for terms like "Jacobi product" I get lots of hits on his triple product formula, and inserting "discriminant" into the search is not helping on isolating a reference that would discuss Jacobi's own work on this.