▁ge odes ic ▁distances ▁between ▁points ▁on ▁the ▁surface ▁independently ▁of ▁the ▁particular ▁way ▁in ▁which ▁the ▁surface ▁is ▁located ▁in ▁the ▁ambient ▁E uclidean ▁space . ▁The ▁crow ning ▁result , ▁the ▁The or ema ▁E greg ium ▁of ▁G auss , ▁established ▁that ▁the ▁Gaussian ▁curv ature ▁is ▁an ▁intr insic ▁invariant , ▁i . e . ▁invariant ▁under ▁local ▁is omet ries . ▁This ▁point ▁of ▁view ▁was ▁extended ▁to ▁higher - dimensional ▁spaces ▁by ▁Riemann ▁and ▁led ▁to ▁what ▁is ▁known ▁today ▁as ▁Riemann ian ▁geometry . ▁The ▁nin ete enth ▁century ▁was ▁the ▁golden ▁age ▁for ▁the ▁theory ▁of ▁surfaces , ▁from ▁both ▁the ▁topological ▁and ▁the ▁differential - ge ometric ▁point ▁of ▁view , ▁with ▁most ▁leading ▁geom eters ▁dev oting ▁themselves ▁to ▁their ▁study . ▁Dar b oux ▁collected ▁many ▁results ▁in ▁his ▁four - volume ▁treat ise ▁Thé orie ▁des ▁surfaces ▁( 1 8 8 7 – 1 8 9 6 ). ▁ ▁The ▁presentation ▁below ▁largely ▁follows ▁G auss , ▁but ▁with ▁important ▁later ▁contributions ▁from ▁other ▁geom eters . ▁For ▁a ▁time ▁G auss ▁was ▁Cart ograph er ▁to ▁George ▁III ▁of ▁Great ▁Britain ▁and ▁Han over ; ▁this ▁royal ▁patron age ▁could ▁explain ▁why ▁these ▁papers ▁contain ▁practical ▁calculations ▁of ▁the ▁curv ature ▁of ▁the ▁earth ▁based ▁purely ▁on ▁measurements ▁on ▁the ▁surface ▁of ▁the ▁planet . ▁ ▁Cur v ature ▁of ▁surfaces ▁in ▁E uclidean ▁space ▁▁ ▁Inform ally ▁G auss ▁defined ▁the ▁curv ature ▁of ▁a ▁surface ▁in ▁terms ▁of