标题： 惠更斯的等时曲线与阿贝尔：从构造几何到分数微积分 显示英文标题

标题： The Tautochrone of Huygens and Abel: From Constructive Geometry to Fractional Calculus

摘要： 在本文中，我们探讨克里斯蒂安·惠更斯和尼尔斯·亨里克·阿贝尔通过等时曲线问题之间的联系。 该问题——确定一条曲线，使得在重力作用下沿该曲线下降的粒子无论起始点如何，所需时间都相同——一直是物理、几何和分析学交叉领域的核心主题。 尽管这两位重要人物相隔近两个世纪，但他们以截然不同的方式解决了这个问题。 虽然惠更斯提出了基于几何构造的物理解决方案，但阿贝尔则在积分方程的分析框架内处理这个问题，采用了一种可以被视为预见并为任意阶微分学的发展铺平道路的方法。 这种对比突显了一个更广泛的历史叙述：数学思维从构造几何向抽象分析的转变。 显示英文摘要

摘要： In this paper, we explore the connections between Christiaan Huygens and Niels Henrik Abel through the tautochrone problem. The problem -- determining the curve along which a particle descends under gravity in the same time, regardless of its starting point -- has been a central topic at the intersection of physics, geometry, and analysis. Though these two major figures are separated by nearly two centuries, they approached the problem in radically different ways. While Huygens proposed a physical solution based on geometric construction, Abel approached the problem within the analytic framework of integral equations, employing a procedure that can be seen as anticipating and paving the way for the development of differential calculus of arbitrary order. This contrast highlights a broader historical narrative: the transformation of mathematical thinking from constructive geometry to abstract analysis.