Diagrams in pre-modern geometry

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K00283
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 01080:Sociology of science, history of science and technology-related
• Research Institution: Yokkaichi University
• Principal Investigator: Saito Ken 四日市大学, 関孝和数学研究所, 研究員 (10221988)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: Apollonius / Conics / Mathematical diagram

Outline of Final Research Achievements

We have redrawn the diagrams of Apollonius' Conics following the manuscript Vat.gr.206, which is the archetype of all the extant Greek manuscripts. In all the diagrams the conic sections are substituted by circle arcs, making this complicated work even more challenging for readers. A program has been developed which aids the user to replace the diagrams by mathematically correct ones with real conic sections.
The overspecification, which is often seen in the diagrams of Euclid's Elements, has proved to be less often in the Conics. This may support the thesis that overspecification in extant manuscripts has ancient origin. However, the Conics may well have escaped overspecification since the extant version has rather late origin (in Eutocius, 6th cent.), and this difficult work does not seem to have been copied often.

Free Research Field

数学史

Academic Significance and Societal Importance of the Research Achievements

古代ギリシャの論証数学は，近現代数学の起源の一つである．本研究が対象としたアポロニオス『円錐曲線論』は現代から見ても高度な内容を含み，しかもガリレオによる投射体の軌跡（放物線），ケプラーによる惑星軌道（楕円）の発見，デカルトによる座標概念の発明の前提となった重要な著作である．本研究は，この著作の伝承過程の一端を明らかにするものである．この難解な著作は，正確な図版を描く器具なしに書写されて，きわめて不正確な図版によって伝承されたにもかかわらず，近世にその内容と議論が正確に復元され，近代数学・物理学が可能になったのである．