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It's a bit like asking who invented the wheel, but perhaps there's something out there. This seems beyond the obvious crew: Euclid, Pythagoras, etc. Is there any evidence when this became common knowledge.

Subquestion: Same thing but for geodesics.

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    $\begingroup$ Crows and bees, perhaps, despite the English language dates later than "the obvious crew" and both the word "beeline" and the expression "as the crow flies" date 18xx. As to homo sapiens, the first reliable evidence of built roads dates about 4000 BC, and those were pretty straight AFAIK. Animal paths and rivers that humans started to use as roads since at least 6000 BC give no indication of whether the travelers understood the idea but I doubt they walked along sine curves to cross the open space. The history of "geodesics" is documented better. Not voting to close yet, but close to it. $\endgroup$ – fedja Aug 30 '13 at 13:34
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    $\begingroup$ The problem is that Euclid, say, probably couldn't give a formal definition of (the length of) an arbitrary curve, but both the triangle inequality and the induction were not unfamiliar to him, so he definitely "realized" that any broken line with the same endpoints is longer. Any ideas how to make the question reasonably unambiguous, at least? $\endgroup$ – fedja Aug 30 '13 at 14:27
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    $\begingroup$ Also the very ideas of "theorem" and "proof" in geometry were introduced in the ancient Greece (presumably by Thales of Miletus), so we are back to the "standard crew". We can trace most modern ideas to the period between Thales (600 BC) and Archimedes (200 BC). Before that there was no "deductive science", after that nobody (except the church) doubted that the power of deductive reasoning is unlimited. Given that this "intellectual revolution" occurred in the slave society on the background of constant wars, the scientific progress of the 20th century doesn't look so impressive any more... $\endgroup$ – fedja Aug 30 '13 at 14:49
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    $\begingroup$ Fedja's comment about crossing open space raise an interesting issue, namely that length-minimization is not the only (or even, in my opinion, the most) plausible explanation for why we cross open spaces along straight lines. An alternative explanation is that we simply walk directly toward our goal. The difference can probably be tested: What do (uneducated) people do when, between them and their goal, there is dry land and also a shallow river, and fording it is slower than walking on land? Efficiency here yields Snell's law, but I expect people will follow straight lines. $\endgroup$ – Andreas Blass Aug 30 '13 at 14:49
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    $\begingroup$ @Andreas Does spending time at the beach count as education? People who spend much time at the beach do approximate Snell's law when they want to go somewhere in the water in a hurry. They run faster on the sand but turn into the water before their target is perpendicular to the shore. They minimize time, so far as they can. $\endgroup$ – Colin McLarty Aug 30 '13 at 15:36
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In order to be a question about mathematics this would have to ask not when the fact became common knowledge -- since it is already known to bees and dogs as mentioned -- but when it got expressed as a theorem. In fact there was a traditional objection to the whole idea of proof in geometry by people who said common sense was better. They cited precisely this fact about shortest paths and found it ridiculous to prove a fact known even to asses. As also noted above, theorems to this effect are about as old as the idea of "theorem" itself, so on one hand the sources are largely lost and on the other hand we should not expect a clear cut answer even if we had all the original sources.

Further, once it becomes mathematics, there is not just one fact about a straight line being the shortest distance. You have to say shortest among what options.

Euclid proves a straight line is shorter than any two other straight lines joining the same ends. He does not offer any more general concept of length of a path. It might be interesting to know who first announced such a fact for every "path" in the plane, with curves in mind, but you can be sure it was before any clear ideas of continuity or smoothness were known, so it cannot be a very good proof by our standards. And it was probably made as an unconscious assumption much more often that stated as a fact.

Carlo Beenaker well points out that a major example of geodesics was known to Menelaus of Alexandria. As to distance Menelaus proves analogues of Euclid's theorems such as the side opposite the greatest angle of a (spherical) triangle is the longest. The rise of the idea of geodesics in general will go notably through Gauss and Riemann, and then the whole idea of "shortest path" gets involved with calculus of variations. There is a huge amount known of this to historians and no doubt more to be found.

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Answering your second question, Menelaus of Alexandria, in his boek Sphaerica, was the first to recognize geodesics on a curved surface as natural analogs of straight lines.

Further reading, here.

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