{"page_number":113,"title":"Page 113","overview":"This page provides a biographical and intellectual overview of Eudoxus of Cnidus, an ancient Greek mathematician, astronomer, and philosopher. It details his education, travels, contributions to observational astronomy, his theory of proportion, the method of exhaustion, and his philosophical views, contrasting them with Plato's.","text_summary":"The text focuses on Eudoxus of Cnidus, highlighting his significant contributions to ancient Greek thought. Eudoxus is credited with advancing observational astronomy and developing the first sophisticated geometrical model of celestial motion. He also wrote on geography and engaged in philosophical discussions at Plato's Academy, though none of his original writings survive. His work is known primarily through later accounts, such as those by Laërtius.\n\nHis education included studying mathematics with Archytas of Tarentum and medicine with Philistion of Locri. He attended lectures in Athens, possibly at Plato's Academy, around 387 BCE. He then traveled to Egypt for 16 months to study with priests, after which he worked as a teacher in Asia Minor before returning to Athens and re-associating with Plato's Academy.\n\nPhilosophically, Aristotle preserved Eudoxus's views on metaphysics, physics, and ethics. Eudoxus believed that forms exist within perceptible things, a view distinct from Plato's. He identified \"the good\" with pleasure, as it is what all things naturally aim for. He eventually returned to his native Cnidus, where he served as a legislator and continued his research until his death at age 53. Notable followers included Menaechmus and Callippus.\n\nIn mathematics, Eudoxus's theory of proportion (equal ratios) laid the groundwork for the general account found in Book V of Euclid's *Elements*. Unlike previous methods that required separate proofs for different types of magnitudes (lines, surfaces, solids), Eudoxus developed general proofs. He is also recognized for formulating the bisection principle, which states that given two magnitudes of the same kind, one can repeatedly cut off at least half of the larger, a principle fundamental to the method of exhaustion. This method, attributed to Eudoxus, was used to determine the volume of a pyramid and a cone, as well as in his work *On the Sphere*.","content_markdown":"# Page 113\n\n### Page Overview\nThis page provides a biographical and intellectual overview of Eudoxus of Cnidus, an ancient Greek mathematician, astronomer, and philosopher. It details his education, travels, contributions to observational astronomy, his theory of proportion, the method of exhaustion, and his philosophical views, contrasting them with Plato's.\n\n### Text Content Summary\nThe text focuses on Eudoxus of Cnidus, highlighting his significant contributions to ancient Greek thought. Eudoxus is credited with advancing observational astronomy and developing the first sophisticated geometrical model of celestial motion. He also wrote on geography and engaged in philosophical discussions at Plato's Academy, though none of his original writings survive. His work is known primarily through later accounts, such as those by Laërtius.\n\nHis education included studying mathematics with Archytas of Tarentum and medicine with Philistion of Locri. He attended lectures in Athens, possibly at Plato's Academy, around 387 BCE. He then traveled to Egypt for 16 months to study with priests, after which he worked as a teacher in Asia Minor before returning to Athens and re-associating with Plato's Academy.\n\nPhilosophically, Aristotle preserved Eudoxus's views on metaphysics, physics, and ethics. Eudoxus believed that forms exist within perceptible things, a view distinct from Plato's. He identified \"the good\" with pleasure, as it is what all things naturally aim for. He eventually returned to his native Cnidus, where he served as a legislator and continued his research until his death at age 53. Notable followers included Menaechmus and Callippus.\n\nIn mathematics, Eudoxus's theory of proportion (equal ratios) laid the groundwork for the general account found in Book V of Euclid's *Elements*. Unlike previous methods that required separate proofs for different types of magnitudes (lines, surfaces, solids), Eudoxus developed general proofs. He is also recognized for formulating the bisection principle, which states that given two magnitudes of the same kind, one can repeatedly cut off at least half of the larger, a principle fundamental to the method of exhaustion. This method, attributed to Eudoxus, was used to determine the volume of a pyramid and a cone, as well as in his work *On the Sphere*.\n\n### Visual Elements (Diagrams, Figures, Graphs, Portraits, Illustrations)\n*No visual elements on this page.*","has_visuals":0,"visual_count":0,"visuals":[]}