The discovery of a gradual acceleration in the moons mean motion by Edmond Halley in the last decade of the seventeenth century led to a revival of interest in reports of astronomical observations from antiquity. These observations provided the only means to study the moons secular acceleration, as this newly-discovered acceleration became known. This book contains the first detailed study of the use of ancient and medieval astronomical observations in order to investigate the moons secular acceleration from its discovery by Halley to the establishment of the magnitude of the acceleration by Richard Dunthorne, Tobias Mayer and Jérôme Lalande in the 1740s and 1750s. Making extensive use of previously unstudied manuscripts, this work shows how different astronomers used the same small body of preserved ancient observations in different ways in their work on the secular acceleration. In addition, this work looks at the wider context of the study of the moons secular acceleration, including its use in debates of biblical chronology, whether the heavens were made up of æther, and the use of astronomy in determining geographical longitude. It also discusses wider issues of the perceptions and knowledge of ancient and medieval astronomy in the early-modern period. This book will be of interest to historians of astronomy, astronomers and historians of the ancient world.