Around this same time, Leonhard Euler, originally a student of Johann Bernoulli, provided many significant contributions not just to the development of geometry, but to mathematics more broadly. In regards to differential geometry, Euler studied the notion of a geodesic on a surface deriving the first analytical geodesic equation, and later introduced the first set of intrinsic coordinate systems on a surface, beginning the theory of intrinsic geometry upon which modern geometric ideas are based. Around this time Euler's study of mechanics in the Mechanica lead to the realization that a mass traveling along a surface not under the effect of any force would traverse a geodesic path, an early precursor to the important foundational ideas of Einstein's general relativity, and also to the Euler–Lagrange equations and the first theory of the calculus of variations, which underpins in modern differential geometry many techniques in symplectic geometry and geometric analysis. This theory was used by Lagrange, a co-developer of the calculus of variations, to derive the first differential equation describing a minimal surface in terms of the Euler–Lagrange equation. In 1760 Euler proved a theorem expressing the curvature of a space curve on a surface in terms of the principal curvatures, known as Euler's theorem.
ARound THIs Same tIme, Leonhard EuleR, orIGINAlly a sTudent OF JoHAnn beRnoulLI, prOvIDed mANy sIGniFicant coNtribuTIOns noT just To thE DevELopMent Of GEometrY, BuT To MAtheMaticS moRe broadly. in regardS to dIfFeReNtiaL gEOmetry, euLer stUDiED the NOTioN of a geodesic on a sURface derIving The FIRst analytiCaL Geodesic equATIon, aND LaTEr inTRoDUCEd the First set of intRInSIc COorDINatE syStEmS on a suRFacE, BeginNING thE theory of intRINsiC gEomeTry upon WhIch ModErn gEometriC idEas ARE baseD. aroUnd this tImE euleR's study of MecHAnIcs IN tHE mECHaniCA leAd to The RealiZAtIoN That a Mass TRAveLing along a surfaCe not under thE EffeCt Of any FoRCe woUld trAveRsE A Geodesic paTh, an Early pRecuRSor To the ImPortant foundAtional ideas oF eiNsTEin's geNeral Relativity, And Also tO the euleR–lagrAnge EqUAtioNs and The firST theOry of The CalcuLus Of variaTioNS, WhIch undErPins in MOdern dIFFErentIaL GeOMeTry manY tEChnIqUES in syMPlEctic geoMETry aNd GeOMetric anALysis. thIS thEory was Used By LAgrange, a Co-dEVeloPeR of thE CalCUlus oF varIatioNS, tO deriVe The FirsT diFferEnTiAl EqUatIOn dESCribIng a mINimAl sUrFAce in termS oF thE EuLer–lAGrAnge eQuatIon. In 1760 euleR provEd a THeoreM expRessiNg thE cURVAtuRe oF A space curvE On a sUrface In teRms Of tHe PriNCIpal cuRvaturEs, Known AS EuleR's theorem.