Editing Differential geometry (section)

=== After calculus (1600–1800) ===
[[File:Osculating circle.svg|thumb|right|An osculating circle of plane curve]]
The first systematic or rigorous treatment of geometry using the theory of infinitesimals and notions from [[calculus]] began around the 1600s when calculus was first developed by [[Gottfried Leibniz]] and [[Isaac Newton]]. At this time, the recent work of [[René Descartes]] introducing [[analytic geometry|analytic coordinates]] to geometry allowed geometric shapes of increasing complexity to be described rigorously. In particular around this time [[Pierre de Fermat]], Newton, and Leibniz began the study of [[plane curve]]s and the investigation of concepts such as points of [[inflection point|inflection]] and circles of [[osculating circle|osculation]], which aid in the measurement of [[curvature]]. Indeed, already in his [[Nova Methodus pro Maximis et Minimis|first paper]] on the foundations of calculus, Leibniz notes that the infinitesimal condition <math>d^2 y = 0</math> indicates the existence of an inflection point. Shortly after this time the [[Bernoulli family|Bernoulli brothers]], [[Jacob Bernoulli|Jacob]] and [[Johann Bernoulli|Johann]] made important early contributions to the use of infinitesimals to study geometry. In lectures by Johann Bernoulli at the time, later collated by [[Guillaume de l'Hôpital|L'Hopital]] into [[Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes|the first textbook on differential calculus]], the tangents to plane curves of various types are computed using the condition <math>dy=0</math>, and similarly points of inflection are calculated.<ref name="struik1" /> At this same time the [[orthogonality]] between the osculating circles of a plane curve and the tangent directions is realised, and the first analytical formula for the radius of an osculating circle, essentially the first analytical formula for the notion of [[curvature]], is written down.

In the wake of the development of analytic geometry and plane curves, [[Alexis Clairaut]] began the study of [[space curve]]s at just the age of 16.<ref>Clairaut, A.C., 1731. Recherches sur les courbes à double courbure. Nyon.</ref><ref name="struik1" /> In his book Clairaut introduced the notion of tangent and [[subtangent]] directions to space curves in relation to the directions which lie along a [[surface]] on which the space curve lies. Thus Clairaut demonstrated an implicit understanding of the [[tangent space]] of a surface and studied this idea using calculus for the first time. Importantly Clairaut introduced the terminology of ''curvature'' and ''double curvature'', essentially the notion of [[principal curvature]]s later studied by Gauss and others.

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.<ref>{{MacTutor|id=Euler|title=Leonhard Euler}}</ref> 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.<ref name="struik1" /> 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 (differential geometry)|Euler's theorem]].

Later in the 1700s, the new French school led by [[Gaspard Monge]] began to make contributions to differential geometry. Monge made important contributions to the theory of plane curves, surfaces, and studied [[surfaces of revolution]] and [[envelope (mathematics)|envelopes]] of plane curves and space curves. Several students of Monge made contributions to this same theory, and for example [[Charles Dupin]] provided a new interpretation of Euler's theorem in terms of the principle curvatures, which is the modern form of the equation.<ref name="struik1" />