The mainstream of the history of logic begins in ancient Greece and comes down through the Arabian and European logic of the Middle Ages and through a number of post-Renaissance thinkers to the more or less mathematical developments in logic in the nineteenth and twentieth centuries. In the period after the fall of Rome many of the ancient achievements were forgotten and had to be relearned; the same thing happened at the end of the Middle Ages. Otherwise this Western tradition has been fairly continuous. Indian and Chinese logic developed separately. Today logic, like other sciences, is studied internationally, and the same problems are treated in the Americas, western and eastern Europe, and Asia and Australasia. The story of the development of logic will be told here under the following headings: