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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"?

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See Anaysis in Ancient Greek Geometry:

What analysis involves is the finding of appropriate principles, previously proved theorems, and constructional moves by means of which the problem can be solved (the desired figure constructed or the relevant theorem proved).

Synthesis instead, as you say, is the "standard" deductive approach: from postulates to theorems.

With Early Modern mathematics, the new algebraic techniques was often called "analytical"; see e.g François Viète and his :

There is a certain way for searching for the truth in mathematics that Plato is said first to have discovered. Theon called it analysis...

Thus, analysis became a sort of catchword for any new mathematical "tools" and methods.

See also:

Regarding Plato, see:

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  • $\begingroup$ Thanks for your answer. Could you give an example of how Plato would have constructed an analytic argument? I need something concrete to understand what it means. $\endgroup$
    – user3339
    Commented Jan 29, 2017 at 1:48
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    $\begingroup$ Related usage (Newton's Method of Fluxions, translated version): "Having observed that most of our modern Geome-- tricians, neglecting the Synthetical Method of the Ancients; have apply'd themselves chiefly to the cultivating of the Analytical Art ; by the assistance of which they have been able to overcome so many and so great difficulties, that they seem to have exhausted all the Speculations of Geometry, ... $\endgroup$
    – Sensebe
    Commented Jan 8, 2019 at 12:42
  • $\begingroup$ …..excepting the Quadrature of Curves, and some other matters of a like nature, not yet intirely discuss'd : I thought it not amiss, for the sake of young Students in this Science, to compose the following Treatise, in which I have endeavour'd to enlarge the Boundaries of Analyticks, and to improve the Doctrine of Curve-lines." (Page 25). $\endgroup$
    – Sensebe
    Commented Jan 8, 2019 at 12:42
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Most problems in Diophantus' Arithmetica are solved in an analytical manner (which is why his style of algebra and its offshoots were described as ars analytica in later times): one starts off assuming a solution exists, and one works one's way towards finding them explicitly. This is why Diophantus has to verify that his solutions indeed satisfy the equation (or "make the problem", as he words it), because he started out with the unwarranted assumption that a solution existed at all.

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