What is algebra and what has it been in history?

Abstract

0. Whoever starts reading Greek mathematics is struck by large parts that are overtly algebraic as well as other parts where algebra seems to hide under a geometrical cover. This fact was a riddle for many for as long as the problems that had beset Greek mathematicians and their solutions were not properly understood. The riddle was solved only after 19th century mathematics had faced the same problems and solved them in a different way. The view that large parts of Greek mathematics are algebraic has never been seriously challenged. Even if there were little left of Greek mathematics, one might conclude from reports on integrals as calculated by Archimedes and from reports on numerical astronomy that the Greeks must have been in possession of powerful algebraic tools. S. Unguru* has recently challenged this view. All who have written about Greek mathematics have been wrong, he claims. On what grounds? Has he discovered sensational new facts? No, nothing! He has not even interpreted old facts in a new way. He simply says they are wrong, and does so with resounding rhetorical emphasis. If the rhetoric is disregarded, the remainder consists of large extracts from the work of others, decorated with numerous exclamation and question marks, and a few, more concise statements, which can properly be submitted to analysis.