This appendix provides mathematical details to supplement the ideas presented in the main text. Topic covered include: angular measurement, apparent diameter, trigonometry, finding the Sun’s altitude from the length of a shadow, determining the relative distances of the Sun and Moon, and finding the distance to an astronomical object using parallax measurements. In addition, this appendix shows how to calculate the sizes of epicycles in the Ptolemaic theory and the periods and sizes of planetary orbits in the Copernican theory. Mathematical details are also provided for Kepler’s Laws of Planetary Motion, Galileo’s measurement of mountains on the Moon, Galileo’s studies of falling bodies and projectiles, Newton’s universal gravitational force, and Bradley’s theory of the aberration of starlight.

In 1671 Robert Hooke thought he had detected an annual parallax for the star Gamma Draconis, thus proving that the Earth orbits the Sun. Setting aside the uncertainty of Hooke’s meagre measurements, there remained the problem of how the Earth could orbit the Sun. Hooke thought he knew: the planets orbited the Sun because of a combination of straight line inertial motion and an attraction toward the Sun. But it was left to Hooke’s rival, Isaac Newton, to work out the mathematical details. While working out these details Newton established an entirely new physics based on three fundamental laws of motion and a universal gravitational attraction between all massive objects. Newton’s physics explained not only the orbits of planets, but also the motion of projectiles, the orbits of the Moon and comets, the precession of the equinoxes, and the tides. Newton’s physics was hailed in England but many European natural philosophers initially dismissed universal gravitational attraction as an “occult quality.”