A Ptolemy diagram is a simple combinatorial object: A collection of diagonals in a polygon with the property that if the collection contains two crossing diagonals, then it also contains all other diagonals connecting end points of the crossing ones.  They could have been defined at any point since antiquity but were in fact introduced recently, in an infinite version, by Puiman Ng.  We give a new combinatorial description of (finite) Ptolemy diagrams, show how they parametrise torsion pairs in cluster categories of Dynkin type A, and count the number of such torsion pairs.