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I am curious, when and by whom it was proved that straight line is the shortest of measurable curves connecting two given points.

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  • $\begingroup$ To me this sounds more like a basic assumption than something that had to be proved. $\endgroup$ – aventurin Jul 21 '16 at 17:31
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    $\begingroup$ There is no way to answer this because it depends on what set of axioms and definitions you use. It could be a definition, or it could be something you prove. $\endgroup$ – Ben Crowell Jul 21 '16 at 19:19
  • $\begingroup$ As pointed out in the above comment by Ben Crowell, it's important to state what kind of framework you're expecting. Modern axioms? Ancient Greece-style mathematics? $\endgroup$ – Danu Jul 21 '16 at 21:51
  • $\begingroup$ Say curves in $\mathbb{R}^n$ for natural $n$, as defined in modern math $\endgroup$ – porton Jul 21 '16 at 22:37
  • $\begingroup$ @porton: "as defined in modern math" doesn't mean anything. $\endgroup$ – Ben Crowell Jul 26 '16 at 16:55
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Essentially Euclid. Exact statement depends on the exact notion of a curve and length. Euclid considers first broken lines, and proves the statement for them. Then he defines the length of other curves (for example a circle) essentially as the limit of lengths of inscribed broken lines. By the way, the modern definition of length is the same. The statement easily follows, because the broken lines can be inscribed in such a way that their length increases.

Remark. Of course, we do not know exactly what Euclid discovered himself, and what was known before him, because his Elements is the only primary source for mathematics before him, and he does not give references.

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    $\begingroup$ Euclid does not do rectification, even of a circle, he only applies the "method of exhaustion" to quadrature. That rectification of a circle reduces to its quadrature was first proved by Archimedes in On the Measurement of the Circle, but even he doesn't define anything like arc length by broken line approximation. That the line segment between two points is shorter than any other path between them is stated as an axiom. ams.org/samplings/feature-column/fc-2012-02 $\endgroup$ – Conifold Jul 25 '16 at 21:55
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    $\begingroup$ Could you tell us just where in the Elements Euclid does all this? $\endgroup$ – Rory Daulton Jul 25 '16 at 22:34
  • $\begingroup$ @Rory Daulton: I am traveling now and have no Euclid with me, but certainly in the place where he discusses the length of a circle, he uses these lemmas. $\endgroup$ – Alexandre Eremenko Jul 28 '16 at 12:04
  • $\begingroup$ Alexandre, please see if mathoverflow.net/questions/152352/is-euclid-dead should be re-opened. $\endgroup$ – Mikhail Katz Aug 17 '16 at 9:28

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