I am trying to understand the difference in "analysis" and "synthesis" as used by the ancient Greek mathematicians. Most sources characterize synthesis as working from givens to a desired conclusion, and analysis as starting from a conclusion and working back to its cause. What would be an example of the Greeks doing this? It seems that all arguments I have read in Euclid's Elements would be synthetic.
Additionally, what is the connection between the historical and modern usages of these terms? Why did calculus come to be called "real analysis"?