  Book 1 develops a mathematical theory of motion under centripetal forces. In keeping with the Euclidean tradition, the propositions mathematically derived from the laws of motion are labeled either as theorems or as problems. The theorems all have an “if-then” form, enabling them to authorize inferences of their consequents, given their  antecedents.[30] But then so too do the problems in effect have an “if-then” logical form, for the (geometric style) solutions they provide authorize inferences from given information to unknowns. The best way to think of the derived propositions, therefore, is as “inference-tickets.” As such, the propositions fall into three categories: (1) ones that license conclusions about forces from information about motions, (2) ones that license conclusions about motions from information about forces, and (3) ones that license conclusions about (net) forces directed toward whole bodies from information about (contributing) forces directed toward the individual parts of the bodies.