Editing Differential geometry (section)

=== Classical antiquity until the Renaissance (300 BC{{Snd}}1600 AD) ===

The study of differential geometry, or at least the study of the geometry of smooth shapes, can be traced back at least to [[classical antiquity]]. In particular, much was known about the geometry of the [[Earth]], a [[spherical geometry]], in the time of the [[ancient Greek]] mathematicians. Famously, [[Eratosthenes]] calculated the [[circumference]] of the Earth around 200 BC, and around 150 AD [[Ptolemy]] in his ''[[Geography (Ptolemy)|Geography]]'' introduced the [[stereographic projection]] for the purposes of mapping the shape of the Earth.<ref name="struik1">Struik, D. J. "Outline of a History of Differential Geometry: I." Isis, vol. 19, no. 1, 1933, pp. 92–120. JSTOR, www.jstor.org/stable/225188.</ref> Implicitly throughout this time principles that form the foundation of differential geometry and calculus were used in [[geodesy]], although in a much simplified form. Namely, as far back as [[Euclid]]'s ''[[Euclid's Elements|Elements]]'' it was understood that a straight line could be defined by its property of providing the shortest distance between two points, and applying this same principle to the surface of the [[Earth]] leads to the conclusion that [[great circles]], which are only locally similar to straight lines in a flat plane, provide the shortest path between two points on the Earth's surface. Indeed, the measurements of distance along such [[geodesic]] paths by Eratosthenes and others can be considered a rudimentary measure of [[arclength]] of curves, a concept which did not see a rigorous definition in terms of calculus until the 1600s.

Around this time there were only minimal overt applications of the theory of [[infinitesimal]]s to the study of geometry, a precursor to the modern calculus-based study of the subject. In [[Euclid]]'s ''[[Euclid's Elements|Elements]]'' the notion of [[tangency]] of a line to a circle is discussed, and [[Archimedes]] applied the [[method of exhaustion]] to compute the areas of smooth shapes such as the [[circle]], and the volumes of smooth three-dimensional solids such as the sphere, cones, and cylinders.<ref name="struik1" />

There was little development in the theory of differential geometry between antiquity and the beginning of the [[Renaissance]]. Before the development of calculus by [[Isaac Newton|Newton]] and [[Leibniz]], the most significant development in the understanding of differential geometry came from [[Gerardus Mercator]]'s development of the [[Mercator projection]] as a way of mapping the Earth. Mercator had an understanding of the advantages and pitfalls of his map design, and in particular was aware of the [[Conformal map projection|conformal]] nature of his projection, as well as the difference between ''praga'', the lines of shortest distance on the Earth, and the ''directio'', the straight line paths on his map. Mercator noted that the praga were ''oblique curvatur'' in this projection.<ref name="struik1" /> This fact reflects the lack of a [[isometry|metric-preserving map]] of the Earth's surface onto a flat plane, a consequence of the later [[Theorema Egregium]] of [[Gauss]].