從刻卜勒到牛頓──分析牛頓的幾何論證

Abstract

本文分析牛頓如何運用幾何方法論證出刻卜勒的三大行星運動定律：透過研讀牛頓的著作《自然哲學之數學原理》，探究牛頓在思考運動力學問題上的幾何思路，並將其觀點以現代數學語言重新呈現。其實牛頓在發展這些理論時，雖然無可避免地要用到極限的概念，但呈現的方式上卻是幾何的。所以本文企圖用常見的幾何論證，來重建牛頓在這一方面的工作，以求科學知識普及化。

In this article, we demonstrate how Newton proved Inverse Square Law of Gravitation through Kepler’s Three Laws of Planetary Motion in geometric ways. By studying Newton’s “Mathematical Principles of Natural Philosophy”, we investigate Newton’s geometric thoughts in mechanics and interpret these thoughts in modern mathematical language. In fact, Newton developed the concepts of Gravitation Theory with Calculus avoidlessly, but presented it in geometric ways. In order to popularize scientific knowledge, I attempt to reestablish Newton’s work in high school geometry.