The axiomatic method begins of course in ancient Greece, and much of the detail of its early evolution is lost to us. By the time of Aristotle it seems to be well established. I mention three aspects of Aristotle's work.

1. The recognition that deduction must begin from unproven principles, and certain terminology relating to these starting points.
2. The recognition that deduction should proceed according to proper rules, and their formalisation in syllogistic logic.
3. The characterisation of a special kind of proof and truth known as "demonstrative"

At this stage, and until quite recently, items 1 and 2 do not fit together. The formal rules of deduction are not adequate or practicable for the kinds of mathematical deduction (notable in geometry) in which the axiomatic method is most successfully applied, and so the axiomatic method is applied only informally, right up to the 20th Century.

In Euclidean geometry is found the most systematic and complete application of the axiomatic method until the twentieth century.