Published by Springer. Copyright 2022 by Mordechai Ben-Ari, This is an open-access book under the Creative Commons Attribution 4.0 International License.
You can download the PDF and order the printed book at SpringerLink.
Figures generated by GeoGebra can also be found at GeoGebra.
This book is intended to give a bird's-eye view of ellipses and planetary orbits. The only background required is secondary-school Euclidean geometry, analytic geometry, and trigonometry.
Although Isaac Newton invented the calculus and used it to study motion, from the time of the Greeks, proof meant proof by geometry. The book contains Newton's detailed geometric proof of the inverse-square law of orbits. Turning to planetary orbits, the book presents Kepler's equation for computing the position, speed and direction of a planet in its orbit, followed by the computation of Lagrange points, which are points in the solar system where a spacecraft can be placed so that the period of its orbit is the same as the Earth's.
Euclid is well-known but mathematicians were equally familiar with Conics by Apollonius of Perga. Some of his results are given in modern notation, although the presentation is faithful to his style. In addition, Kepler's own geometric proof of his First Law is given. The final chapters present challenging theorems on ellipses: the Steiner inellipse, Marden's Theorem, the theorems of Pascal and Brianchon, and Newton's Ellipse Theorem.