Development of the mathematics resembles usually a growing tree - from old branches grow new ones. However sometimes domains of mathematics which were separated for the long time are fused together at one point starting a new domain.

Surely the most famous example (and probably the most fruitful) is the fusion of geometry and algebra into analytic geometry made by Descartes.

Are there also other known fusions in history of mathematical domains which merged into one subject after many years of distinct development?

If it is known to MSE users please give some examples with the most appropriate date of fusion and specific event which could be regarded as the starting point (as Descartes' La Géométrie 1637) and approximate years of separate developments...

Probably it is hard to find other example when such unconnected mathematical developmental processes were so long as in the case of analytic geometry.. but anyway it's interesting what fusion could take the second and the third place in this ranking counting the years of independent development?

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    $\begingroup$ Maybe this qualifies- Modular forms and elliptic curves were thought to be two different domains before Tanyama-Shimura, Andrew Wiles made a connection and solved Fermat's last theorem. $\endgroup$ – Kushal Bhuyan Jul 21 '16 at 11:00
  • $\begingroup$ @Kushal Could you give, Kushal, some specific dates? I wonder how long they were separated.. $\endgroup$ – Widawensen Jul 21 '16 at 11:10
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    $\begingroup$ Taniyama conjectured it (incompletely) around 1955, and improved by Shimura and Weil. Then Andrew Wiles proved the conjecture in 1993, after a 7 year long struggle. Modular forms are in development throughout the 19th and 20th century in phases according to Wikipedia. $\endgroup$ – Kushal Bhuyan Jul 21 '16 at 11:16
  • $\begingroup$ So it is quite long.. has the merged domain a distinct name? $\endgroup$ – Widawensen Jul 21 '16 at 11:20
  • $\begingroup$ I don't think so. People earlier thought that elliptic curves and modular forms has no relation at all. But by proving Taniyama -Shimura conjecture, Andrew Wiles showed that there is a link between these two seemingly different areas. $\endgroup$ – Kushal Bhuyan Jul 21 '16 at 11:25

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