▁is omet ries ▁is ▁a ▁surface ▁of ▁constant ▁curv ature . ▁ ▁A ▁complex ▁surface ▁is ▁a ▁complex ▁two - man if old ▁and ▁thus ▁a ▁real ▁four - man if old ; ▁it ▁is ▁not ▁a ▁surface ▁in ▁the ▁sense ▁of ▁this ▁article . ▁Ne ither ▁are ▁algebraic ▁curves ▁or ▁surfaces ▁defined ▁over ▁fields ▁other ▁than ▁the ▁complex ▁numbers . ▁ ▁History ▁of ▁surfaces ▁ ▁Is ol ated ▁properties ▁of ▁surfaces ▁of ▁revolution ▁were ▁known ▁already ▁to ▁Arch im edes . ▁The ▁development ▁of ▁calculus ▁in ▁the ▁sevent e enth ▁century ▁provided ▁a ▁more ▁system atic ▁way ▁of ▁prov ing ▁them . ▁Cur v ature ▁of ▁general ▁surfaces ▁was ▁first ▁studied ▁by ▁E uler . ▁In ▁ 1 7 6 0 ▁he ▁proved ▁a ▁formula ▁for ▁the ▁curv ature ▁of ▁a ▁plane ▁section ▁of ▁a ▁surface ▁and ▁in ▁ 1 7 7 1 ▁he ▁considered ▁surfaces ▁represented ▁in ▁a ▁paramet ric ▁form . ▁Mon ge ▁laid ▁down ▁the ▁found ations ▁of ▁their ▁theory ▁in ▁his ▁classical ▁mem oir ▁L ' application ▁de ▁l ' analy se ▁à ▁la ▁gé omet rie ▁which ▁appeared ▁in ▁ 1 7 9 5 . ▁The ▁defining ▁contribution ▁to ▁the ▁theory ▁of ▁surfaces ▁was ▁made ▁by ▁G auss ▁in ▁two ▁remarkable ▁papers ▁written ▁in ▁ 1 8 2 5 ▁and ▁ 1 8 2 7 . ▁This ▁marked ▁a ▁new ▁departure ▁from ▁tradition ▁because ▁for ▁the ▁first ▁time ▁G auss ▁considered ▁the ▁intr insic ▁geometry ▁of ▁a ▁surface , ▁the ▁properties ▁which ▁are ▁determined ▁only ▁by ▁the