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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.

Edit: In an article by Hurwitz in 1881 (see here), the product for $\Delta$ appears on p. 552 as a homogeneous function $\Delta(\omega_1,\omega_2)$ of two complex variables $\omega_1$ and $\omega_2$: $(2\pi/\omega_2)^{12} q^2\prod_{k \geq 1} (1 - q^{2k})^{24}$, where his $q$ is $e^{\pi i\omega_1/\omega_2}$. Earlier, Dedekind in 1877 here introduced the related eta-function $\eta(\omega)$: a definition and its transformation under ${\rm SL}_2(\mathbf Z)$ are equations (3) and (6) on p. 281 and its product representation $q^{1/12}\prod_{\nu \geq 1} (1 - q^{2\nu})$ is equation (24) on p. 285, with $q = e^{\pi i\omega}$. Dedekind's paper never mentions $\Delta$.

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.

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.

Edit: In an article by Hurwitz in 1881 (see here), the product for $\Delta$ appears on p. 552 as a homogeneous function $\Delta(\omega_1,\omega_2)$ of two complex variables $\omega_1$ and $\omega_2$: $(2\pi/\omega_2)^{12} q^2\prod_{k \geq 1} (1 - q^{2k})^{24}$, where his $q$ is $e^{\pi i\omega_1/\omega_2}$. Earlier, Dedekind in 1877 here introduced the related eta-function $\eta(\omega)$: a definition and its transformation under ${\rm SL}_2(\mathbf Z)$ are equations (3) and (6) on p. 281 and its product representation $q^{1/12}\prod_{\nu \geq 1} (1 - q^{2\nu})$ is equation (24) on p. 285, with $q = e^{\pi i\omega}$.

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.

Edit: In an article by Hurwitz in 1881 (see here), the product for $\Delta$ appears on p. 552 as a homogeneous function $\Delta(\omega_1,\omega_2)$ of two complex variables $\omega_1$ and $\omega_2$: $(2\pi/\omega_2)^{12} q^2\prod_{k \geq 1} (1 - q^{2k})^{24}$, where his $q$ is $e^{\pi i\omega_1/\omega_2}$. Earlier, Dedekind in 1877 here introduced the related eta-function $\eta(\omega)$: a definition and its transformation under ${\rm SL}_2(\mathbf Z)$ are equations (3) and (6) on p. 281 and its product representation $q^{1/12}\prod_{\nu \geq 1} (1 - q^{2\nu})$ is equation (24) on p. 285, with $q = e^{\pi i\omega}$. Dedekind's paper never mentions $\Delta$.

When did Jacobi prove the product formula for the discriminant function: $\Delta(\tau) = (2\pi)^{12}q\prod_{n \geq 1} (1 - q^n)^{24}$?

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.

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.

Edit: In an article by Hurwitz in 1881 (see here), the product for $\Delta$ appears on p. 552 as a homogeneous function $\Delta(\omega_1,\omega_2)$ of two complex variables $\omega_1$ and $\omega_2$: $(2\pi/\omega_2)^{12} q^2\prod_{k \geq 1} (1 - q^{2k})^{24}$, where his $q$ is $e^{\pi i\omega_1/\omega_2}$. Earlier, Dedekind in 1877 here introduced the related eta-function $\eta(\omega)$: a definition and its transformation under ${\rm SL}_2(\mathbf Z)$ are equations (3) and (6) on p. 281 and its product representation $q^{1/12}\prod_{\nu \geq 1} (1 - q^{2\nu})$ is equation (24) on p. 285, with $q = e^{\pi i\omega}$.