Longomontanus’ De maculis in Luna and the determination of terrestrial longitudes

In Astronomia Danica, Longomontanus provides a method for calculating the terrestrial longitude for a given location on Earth. To do so, he relies on precise calculations of the lunar position so that he can know when he is observing it without any parallax in longitude. As I will show, this method has a fatal flaw that renders it unusable. However, Longomontanus also provides a simple observational method that indicates, via the disposition of the lunar spot and/or horns, the time when the Moon shows no parallax in longitude. This last method, though it does not need any use of tables, also has some problems. In this paper I will explain in detail these methods provided by Longomontanus, together with the problems they carry.