  The quadratrix received its name from the circle quadrature, squaring the
circle, but it only solves it if completed by taking a limit, as pointed out
already in antiquity. We ask if it can square the circle without limits and
restrict its use accordingly, to converting ratios of angles and segments into
each other. The problem is then translated into algebra by analogy to
straightedge and compass constructions, and leads to an open question in
transcendental number theory. In particular, Lindemann's impossibility result
no longer suffices, and the answer depends on whether $\pi$ belongs to the
analog of Ritt's exponential-logarithmic field with an algebraic base. We then
derive that it does not from the well-known Schanuel conjecture. Thus, the
quadratrix so restricted cannot square the circle after all.
