Tentative Reconstruction of the History of the Arithmetica

In the beginning of the seventh Book of his Collection, Pappus mentions two types of analyses and syntheses distinguished by the Greeks.1 The first, ποριστικóν, type is commonly used by geometers in connection with the demonstration of a proposition, or of an (already known) solution. In the corresponding analysis, what is to be proved is supposed to be true (or known), and must be reduced by passing through its successive consequences, either to an identity or to a known proposition. The synthesis then reverses the process. The second kind of analysis, of the ζητητικóν type, is used in the finding of a solution to a problem. Supposing the problem solved, the mathematician establishes between the known and the unknown magnitudes some relation, which is then reduced, by elimination, to a final relation containing the smallest number of unknowns possible (one for a determinate problem). This is the analysis. The synthesis simply verifies the exactness of the solution found.