## The Ballet of the Planets: A Mathematician's Musings on the Elegance of Planetary MotionThe Ballet of the Planets unravels the beautiful mystery of planetary motion, revealing how our understanding of astronomy evolved from Archimedes and Ptolemy to Copernicus, Kepler, and Newton. Mathematician Donald Benson shows that ancient theories of planetary motion were based on the assumptions that the Earth was the center of the universe and the planets moved in a uniform circular motion. Since ancient astronomers noted that occasionally a planet would exhibit retrograde motion—would seem to reverse its direction and move briefly westward—they concluded that the planets moved in epicyclic curves, circles with smaller interior loops, similar to the patterns of a child's Spirograph. With the coming of the Copernican revolution, the retrograde motion was seen to be apparent rather than real, leading to the idea that the planets moved in ellipses. This laid the ground for Newton's great achievement—integrating the concepts of astronomy and mechanics—which revealed not only how the planets moved, but also why. Throughout, Benson focuses on naked-eye astronomy, which makes it easy for the novice to grasp the work of these pioneers of astronomy. |

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#### The Ballet of the Planets: A Mathematician's Musings on the Elegance of Planetary Motion

User Review - Ian D. Gordon - Book VerdictBenson (mathematics, emeritus, Univ. of California, Davis; A Smoother Pebble: Mathematical Explorations) takes readers on a historical journey through the development of mathematics and geometric ... Read full review

### Contents

1 The Survival of the Valid | 3 |

Birth | 9 |

Rebirth | 77 |

Enlightenment | 119 |

A The Greek Alphabet | 163 |

B Vectors | 165 |

Notes | 167 |

173 | |

175 | |

### Common terms and phrases

analemma ancient astronomers angle angular momentum angular velocity Archimedes axis celestial sphere center of mass chapter clockwise concept conﬁrmed constant Copernican Copernicus Coriolis effect deferent and epicycle deferent circle deferent-epicycle model deﬁned deﬁnition difﬁcult discussed distance Earth Earth’s orbit Earthship ellipse elliptical orbit epicyclic curves epicyclic motion equal equant example ﬁgure ﬁnd ﬁrst ﬁt ﬁxed stars force Galileo geocentric geometry heliocentric theory inferior planets initial intersection inverse-square law Kepler Kepler’s third law latitude line segment linkage Martian mathematical measured mechanics merry-go-round modiﬁcation motion of Mars motionless moves Newton observations ofMars ofthe parallax particle path planetary motion point Q position problem Proposition Ptolemy Ptolemy’s radius ratio respectively retrograde loops retrograde motion right ascension rods rotation rate scientiﬁc second law Section semimajor axis shown in Figure shows solar system speciﬁed speed sufﬁcient superior planets tangent theorem triangle two-circle epicyclic curve Tycho uniform circular motion vector Venus