[aap_]

Square roots of negative numbers came up when solving cubic equations, even if the final solutions were all real. This meant the square root of a negative number was not something nonsensical the way you might claim for x^2 = -1, but actually...real in some sense.

[caf]

Specifically I believe it involved a geometric construction for solving the cubics, which in some cases could not find a solution unless you allowed a square with "negative area".