Origin of 'Analytic' Geometry? My impression is that the name analytic geometry, which I understand roughly to be geometry in Euclidean space using coordinates, is not used that much anymore. We would probably classify the subject as an elementary version of real algebraic geometry these days, even though it's often absorbed into a course on multi-variable calculus. My question is, who coined the term 'analytic geometry'? And what was the sense in which they were using the word 'analytic'? If you know, it would be useful to have some detail on the meaning of the word in this mathematical context rather than philosophical generalities on the 'analytic-synthetic distinction.'
 A: Analysis means breaking apart; taking something complex and decomposing it into simpler constituents.
This is associated with "working backwards": starting with a complicated result and finding simpler ones from which it follows. The Greeks used "analysis" in this sense in mathematics. In this process one assumes a sought result as if it was given, and works "backwards" to uncover from which simpler things it can be derived, with the intention of then reversing the steps to give a direct synthetic (synthesis = putting together) proof of the sought result.
In the 17th century, "analysis" came to mean "working with x" so to speak, because when we call a sought quantity x and start manipulating it in equations then we are indeed treating the sought as if it was known, which is exactly the classical meaning of analysis.
With the advent of calculus, since "analysis" meant "working with x" it also became associated with "working with f(x)", and hence we get analysis in today's sense of real analysis.
