Title: Can the quadratrix truly square the circle? Show Chinese title

Title: 四边形真的能求出圆的面积吗？

Abstract: 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. Show Chinese abstract

Abstract: 曲线名称来源于圆的化方，即化圆为方，但它只有在通过取极限完成时才能解决这个问题，这一点早在古代就已经指出。 我们询问它是否可以在不使用极限的情况下化圆为方，并相应地限制其使用，以将角度和线段的比例相互转换。 然后，该问题通过类比直尺和圆规作图转化为代数问题，并导致超越数理论中的一个开放性问题。 特别是，林德曼的不可能性结果已不再足够，答案取决于$\pi$是否属于具有代数底数的里特指数-对数域的类比。 然后我们从著名的沙努埃尔猜想推导出它不属于该域。 因此，这样限制后的曲线实际上无法化圆为方。