care and precision have been possible, according to the techniques of observation available at any time. There is a correspondingly long history of lunar theories: it stretches from the times of the Babylonian and Greek astronomers, down to modern lunar laser ranging. Among notable astronomers and mathematicians down the ages, whose names are associated with lunar theories, are: Babylonian/Chaldean Naburimannu Kidinnu Soudines Greek/Hellenistic Hipparchus Ptolemy Arab Ibn al-Shatir European, 16th to early 20th centuries Tycho Brahe Johannes Kepler Jeremiah Horrocks Ismaël Bullialdus John Flamsteed Isaac Newton Edmond Halley Leonhard Euler Alexis Clairaut Jean d'Alembert Tobias Mayer Johann Tobias Bürg Pierre-Simon Laplace Philippe le Doulcet Johann Karl Burckhardt Peter Andreas Hansen Charles-Eugène Delaunay John Couch Adams North American, 19th to early 20th centuries Simon Newcomb George William Hill Ernest William Brown Wallace John Eckert Other notable mathematicians and mathematical astronomers also made significant contributions. The history can be considered to fall into three parts: from ancient times to Newton; the period of classical (Newtonian) physics; and modern developments. Ancient times to Newton Babylon Of Babylonian astronomy, practically nothing was known to historians of science before the 1880s. Surviving ancient writings of Pliny had made bare mention of three astronomical schools in Mesopotamia – at Babylon, Uruk, and 'Hipparenum' (possibly 'Sippar'). But definite modern knowledge of any details only began when Joseph Epping deciphered cuneiform texts on clay tablets from a Babylonian archive: In these texts he identified an ephemeris of positions of the Moon. Since then, knowledge of the subject, still fragmentary, has had to be built up by painstaking analysis of deciphered texts, mainly in numerical form, on tablets from Babylon and Uruk (no trace has yet been found of anything from the third school mentioned by Pliny). To the Babylonian astronomer Kidinnu (in Greek or Latin, Kidenas or Cidenas) has been attributed the invention (5th or 4th century BC) of what is now called "System B" for predicting the position of the moon, taking account that the moon continually changes its speed along its path relative to the background of fixed stars. This system involved calculating daily stepwise changes of lunar speed, up or down, with a minimum and a maximum approximately each month. The basis of these systems appears to have been arithmetical rather than geometrical, but they did approximately account for the main lunar inequality now known as the equation of the center. The Babylonians kept very