If $(u,v)$ are isothermal coordinates, then $u$ and $v$ are harmonic functions with respect to the Laplace-Beltrami operator on your Riemannian manifold, that is $\Delta u=\Delta v=0$.
Now, the equation $\Delta f=0$ characterizes the stationary states for the heat equation. The level curves for a harmonic function are therefore the isothermal curves for some heat distribution.
It follows that if $(u,v)$ is an isothermal system of coordinates, then the level curves, i.e., the coordinate lines, are isothermal curves.
A formula was found for expressing solutions of third-degree algebraic equations with real coefficients in terms of addition, subtraction, multiplication, division, and square and cube roots. But in some cases, the number whose square root was to be found was negative. But these paradoxical quantities canceled out, leaving a real number. And now the strange part: when such solutions were substituted into the equation, they checked! That was a reason to pay attention to them.
But they did not arise as quantities in geometry (lengths, areas, etc.) nor in accounting (which is where negative numbers came from), so they were not "real".
(I don't think the concept of "real number" existed until fairly late in the game---some time after all this was done.)
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