牛顿对三次曲线分类的最初研究

Newton’s Original Research on the Classification of Cubic Curves

摘要: 牛顿对三次曲线的分类研究是高次曲线研究领域的奠基性工作。现有研究多聚焦于牛顿在1704年发表的《三次曲线枚举》，本文聚焦于牛顿1667~1668年对三次曲线的最初研究，以牛顿拉丁语手稿及其英译版为基础，旨在厘清牛顿的分类逻辑。作者通过案例分析表明，牛顿先通过坐标变换将三次曲线一般方程化简为9类，再依据渐近线交点、曲线直径是否存在、方程根的情况等几何与代数特征，构建“类–子类–亚类–型”四层分类框架，最终确定了58个不同型的三次曲线。本研究明确了牛顿对三次曲线分类框架的具体标准和技术路线，有助于理解牛顿的最初的分类思想与逻辑。

Abstract: Newton’s research on the classification of cubic curves is a foundational work in the field of higher-order algebraic curve studies. Existing studies have mostly focused on Enumeratio Linearum Tertii Ordinis published by Newton in 1704, while this paper centers on his preliminary research on cubic curves carried out during 1667~1668. Based on Newton’s Latin manuscripts and their English translations, this study aims to clarify the logical system of his classification for cubic curves. Through case analysis, the authors demonstrate that Newton first simplified the general equation of cubic curves into nine categories by means of coordinate transformation, and then constructed a four-tier classification framework of “Class-Subclass-Subsubclass-Type” according to geometric and algebraic characteristics including the intersection points of asymptotes, the existence of curve diameters and the properties of equation roots, ultimately defining 58 distinct types of cubic curves. This study clarifies the specific criteria and technical approach of Newton’s classification framework for cubic curves, which is conducive to an in-depth understanding of his original ideological connotation and logical reasoning regarding the classification of cubic curves.