Scholarlink

In this chapter the development of synthetic geometry, along almost one century and a half, from Pascal’s death to Monge’s invention of Descriptive geometry, is treated. This period saw the contributions to the discipline of several geometers; some tried to divine the lost books of Apollonius’s Conics, others tried to develop systematically the theory of conic sections starting from a single fundamental result. The chapter focus in particular on those due to Borelli, de La Hire, L’Hôspital, Le Poivre, Newton, Maclaurin, Simson, and Boscovich. The first two, on the basis of the projection method and the principle of harmonic division, developed the theory of conics in their works, respectively in Elementa conica, and in Nouvelle méthode etc. later perfected in the treatise Sectiones Conicae. These works are described and commented here in some detail. The second two approached the synthetic theory of conics by putting at its basis the so-called chords theorem (Apollonius’s III, Prop. 16–23) of which they gave a nice proof in space via the simple assumption of the projection method. The geometrical results expounded in Newton’s Principia Mathematica, and what he developed assuming the chord theorem as given in Apollonius’s Conics, are also treated in the chapter. Particular emphasis is given to his solution of the four lines problem of Pappus, that Newton seemed to contrapose to the analytic one already offered by Descartes in his Géometrie; and to the organic construction of conic sections. Maclaurin and Simson provided some theorems of projective character; in particular, they re-discovered the hexagon theorem, and the first also discovered the converse of this theorem, which was independently also obtained by Braikenridge, and for this reason it is today known as the theorem of Braikenridge and Macalurin. Also this story is told in detail in this chapter. Finally, space is given in this chapter to the innovative ideas of Boscovich, who, in particular, introduced explicitely the “Principle of continuity” in geometry.

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

An Interlude a Century and a Half Long

  • 章节

  • 发表日期：

    2025年

查看全文
  • Andrea Del Centina,
  • Alessandro Gimigliano

摘要

In this chapter the development of synthetic geometry, along almost one century and a half, from Pascal’s death to Monge’s invention of Descriptive geometry, is treated. This period saw the contributions to the discipline of several geometers; some tried to divine the lost books of Apollonius’s Conics, others tried to develop systematically the theory of conic sections starting from a single fundamental result. The chapter focus in particular on those due to Borelli, de La Hire, L’Hôspital, Le Poivre, Newton, Maclaurin, Simson, and Boscovich. The first two, on the basis of the projection method and the principle of harmonic division, developed the theory of conics in their works, respectively in Elementa conica, and in Nouvelle méthode etc. later perfected in the treatise Sectiones Conicae. These works are described and commented here in some detail. The second two approached the synthetic theory of conics by putting at its basis the so-called chords theorem (Apollonius’s III, Prop. 16–23) of which they gave a nice proof in space via the simple assumption of the projection method. The geometrical results expounded in Newton’s Principia Mathematica, and what he developed assuming the chord theorem as given in Apollonius’s Conics, are also treated in the chapter. Particular emphasis is given to his solution of the four lines problem of Pappus, that Newton seemed to contrapose to the analytic one already offered by Descartes in his Géometrie; and to the organic construction of conic sections. Maclaurin and Simson provided some theorems of projective character; in particular, they re-discovered the hexagon theorem, and the first also discovered the converse of this theorem, which was independently also obtained by Braikenridge, and for this reason it is today known as the theorem of Braikenridge and Macalurin. Also this story is told in detail in this chapter. Finally, space is given in this chapter to the innovative ideas of Boscovich, who, in particular, introduced explicitely the “Principle of continuity” in geometry.