Ramanujan in his 1912 article Note on a set of simultaneous equations explicitly solves a special system of polynomial equations $\boldsymbol{Y}\boldsymbol{x}=\boldsymbol{a};\ \boldsymbol{Y}\in\mathbb{R}^{2n\times n}\ \boldsymbol{x},\boldsymbol{a}\,\in\,\mathbb{R}^n$; $Y_{ij}=y_j^i$ by introducing a variable $\theta$ that seems to come out of the blue.

My impression from reading the article is that $\theta$ only serves the purpose of identifying corresponding terms after having carried out calculations.