阿波羅尼斯、牛頓和丹德林的橢圓

Abstract

本論文共三章：第一章說明古希臘時期數學家阿波羅尼斯對於圓錐曲線所做的探討，其中一個重要的結論就是橢圓的共軛直徑基本定理。第二章解釋牛頓如何利用共軛直徑基本定理，證明出「向心力和距離平方成反比」此一命題。第三章介紹比利時數學家丹德林提出的圓錐模型，呈現出橢圓焦點和準線在圓錐模型中的意義，我們因此可以計算出圓錐曲線的離心率，進而了解日晷晷影的軌跡。

This thesis consists of three chapters. In chapter one, we review the work of conic sections by Greek mathematician Apollonius, and the theorem on conjugate diameters of ellipse. In chapter two, we explain Newton’s proof of “the centripetal force is inversely proportional to the square of distance”. In the proof, the above mentioned theorem on conjugate diameters played an important role. In chapter three, we introduce the conic modeling by Dandelin, a Belgian mathematician, which shows the geometrical meaning of the focus and the directrix of ellipse. We then calculate the eccentricity of conic sections by Dandelin’s modeling and classify the locus of the sundial shadow.