非欧几何学早期历史的研究与普及

The early development of non-Euclidean geometry and the accompanying revolution in geometrical thoughts fascinate researchers in the field of history of mathematics. Previous studies in this topic focus on the division of historical stages, the contributions of mathematicians, and questions about priority. From the viewpoint of “why did these breakthroughs in each stage occur?”, we put forward the following historical issues: First, why did the study on parallel postulate has no progress for about 2000 years until Saccheri made a breakthrough? Second, why did it take six decades to construct the first non-Euclidean geometry corresponding to the acute angle assumption, which was proposed by Saccheri in 1733? Third, why didn’t Lobachevskii and Bolyai’s original works on non-Euclidean geometry obtain widespread concern until Beltrami reinterpreted them? We will investigate the essence of these breakthroughs in each stage, and how they are related to the development of mathematical instruments and methods in the 18th and 19th century. In particular, the development of analysis and differential geometry played a vital role in the founding and acceptance of non-Euclidean geometry. We will investigate these issues in deep and obtain a thorough landscape of the early history of non-Euclidean geometry. Also, we will popularize the historical knowledge of non-Euclidean geometry based on our study.

非欧几何学的早期发展，以及与之相伴的几何学思想的革新，是国内外数学史界研究的热点。以往关注的重点是非欧几何学历史阶段的划分、数学家贡献的总结和优先权的讨论。本项目从“为什么这些阶段性突破会出现”的角度，提出下列议题：一、为什么平行公设的研究持续了两千年之久没有进展，而到Saccheri的时期可以有所进步？二、Saccheri1733年提出三种几何假设，为什么到1830年前后才出现与锐角假设对应的罗氏几何？三、为什么Lobachevskii和Bolyai提出的非欧几何没能直接引起反响，直到1868年Beltrami解读非欧几何的工作发表之后非欧几何才获得广泛的认可?本项目将研究各阶段性突破同当时的背景以及数学工具的关系，特别是分析学与微分几何学工具在非欧几何建立与接受过程中的作用。基于以上研究，我们将系统而深入地呈现非欧几何学建立与接受过程的整体图景，并试图在普及非欧几何学历史方面有所贡献。

本项目致力于研究和普及非欧几何学的早期历史。通过分析平行公设问题的历史起源，梳理非欧几何学的历史脉络与阶段划分，以及研究非欧几何学历史上关键突破发生的原因，理解和呈现非欧几何学的历史图景，并为普及非欧几何学的历史做出努力。项目完成译著《空间的思想：欧式几何，非欧几何，与相对论》的书稿，签订翻译出版合同并且即将出版，将弥补国内没有非欧几何学史经典专著中文译本的空白。发表科普文章《古希腊演绎数学的起源》、《非欧几何学的历史与阶段划分》、《指南车：来自微分几何学的邀请》等，将有助于普及与传播非欧几何学的历史，并为数学文化、数学教育等领域提供相关素材。