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Who and when - simple piece of geometry

  1. Dec 17, 2015 #1

    julian

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    It is well known that:

    The shortest distance from a point to a line is the length of the line segment which is perpendicular to the line and joins to the point.

    Who first proved this? How far back in time does it go?
     
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  3. Dec 17, 2015 #2

    DaveC426913

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    That's part of Euclidean metric. I'm not sure he first proved it though.
     
  4. Dec 17, 2015 #3
    Euclid, 300 BC was the first to publish the result, so he gets the credit.

    As for the shortest distance between two points being a straight line, Euclid quoth, "Any ass knows this."
     
  5. Dec 17, 2015 #4

    julian

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    I'm only asking about "The shortest distance from a point to a LINE".
     
  6. Dec 17, 2015 #5
    I'm not sure it's true Euclid proved anything about the "shortest possible distance." Here is the proposition in question:

    http://aleph0.clarku.edu/~djoyce/elements/bookI/propI12.html

    I can't find any claim made, or proof of the idea, prior to this that a straight line is the shortest possible distance between two points, though, and that claim is not made in this proposition, 12.

    Maybe someone knows where in Elements this specific claim is proved, if anywhere.
     
  7. Dec 17, 2015 #6

    DaveC426913

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  8. Dec 17, 2015 #7
    Wikipedia has some proofs of it here:

    http://aleph0.clarku.edu/~djoyce/elements/bookI/propI12.html

    But no one is credited with these proofs and there's no mention of the date of their origins. They seem modern to me.

    This site:

    http://geomalgorithms.com/a02-_lines.html

    has this comment:
    which characterizes this idea as "intuitive," something that suggests the greeks felt no urgency to construct a proof.
     
  9. Dec 17, 2015 #8

    Yes I know. I was trying to show something amusing, not insult you. Sorry.
     
  10. Dec 18, 2015 #9
    Some would say that the first to rigorously prove anything in geometry was Hilbert, who more or less rewrote Euclid in the early 1900's.
     
  11. Dec 19, 2015 #10

    epenguin

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    I will take advantage of the fact that this is not the professional mathematicians' section of the forum to communicate that after deep meditation for 24 hours I concluded that this cannot be a question about lines, since Euclid never defined them, making it difficult to deduce anything about them, it's about distances, which I think he did - I think it's about the number of times one segment of a line fits into another of the same or a different line. Well that's what I would've done. I think I could prove the theorem in question to my, I don't say your, satisfaction. :-p
     
  12. Dec 19, 2015 #11

    DaveC426913

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    Well, yes. Of the four posters to this thread, each has mentioned distances explicitly.

    Not sure I follow.
     
  13. Dec 19, 2015 #12
    http://aleph0.clarku.edu/~djoyce/elements/bookI/bookI.html

    He does define lines. However, he does not define straight lines as, or prove them to be, the shortest distance between two points. You would need to prove that before proving they are the shortest distance between a line and a point not on that line.
     
  14. Dec 19, 2015 #13

    epenguin

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    Ah so he does. But I thought that definition was regarded now as not properly mathematical, and that a line was an thing defined by two points?

    What makes you think you're meant to? It Can Be Shown I tell you! :oldbiggrin:
     
  15. Dec 19, 2015 #14
    Perhaps that's true, but there is a difference between saying, "...Euclid never defined them...," and saying, "that definition was regarded now as not properly mathematical." Euclid (or whoever) certainly made a serious attempt at a rigorous definition.

    I am not sure where the definitions in Elements might start to be seen as inadequate, but I'd suspect it's probably not till non-Euclidian geometry. If that's the case, I wonder if it's of any benefit to go there, since the OP's question seemed to be perfectly classical.
     
  16. Dec 19, 2015 #15

    epenguin

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    I don't know how that 'definition' I now recall perfectly well eluded me. I think it had been devalued from hearing it declared worthless. I have to say the first time I hear it I thought that 'that which', an unusual English expression found only in equally questionable definitions like of force, was an evasion.

    Does it have any operational role in proving anything? I mean anything mathematically significant, I guess with it you can prove that an elephant is not a line.
     
  17. Dec 19, 2015 #16
    Where are you seeing "that which?" I don't see those two words paired in the quote I posted. Regardless, I don't think there is anything intrinsically evasive about using "that which." It would depend on the context.
    If you can use it (the definition of a straight line) to prove anything is not a straight line, then you might have what you need to prove a straight line is the shortest distance from a line to a point not on that line. Things can be proved "in negativa", meaning you prove a thing is true by proving all the possible alternative can't be true. The example I like is how Archimedes proved the center of gravity of a triangle must lie on a line from the some vertex of the triangle to the midpoint of the opposite side.

    Regardless, in looking that proof up, I stumbled across a place in Archimedes where he states as an assumption that a straight line is the shortest distance between two points:

    That's from On the Sphere and the Cylinder, book 1, Assumptions

    As the alternative to 'straight' lines he cites 'bent' lines. So, if we have two lines, a straight one and a bent one, with the same endpoints, the straight line will be the shorter.

    For if it is not, I would add, we should be able to find some 'bent' line with the same endpoints as some straight line which is shorter than that straight line. Which is absurd.
     
  18. Dec 20, 2015 #17

    epenguin

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    In answer to Dave Wikipedia puts it: "the distance of scale is relative; one arbitrarily picks a line segment with a set and non-zero laying as the unit, end of the distances are expressed in relation to it. Addition of distances is represented by a construction in which one line segment is copied onto the end of another line segment to extend its length, and similarly for subtraction."
     
  19. Dec 20, 2015 #18

    epenguin

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    In answer to the zooby, sorry I was quoting the definition I remembered, But the memory it Is very strong, The definition striking, I'm not that much different from yours that you give, which sounds to me similarly evasive.

    I mean isn't he trying to say the straight line is some sort of a thing which is like some other thing that you are familiar with? But in truth it's not a thing at all. At cost of probably telling you what you (but not we all) already know Coxeter says "Euclid did not specify his primitive concepts and relations, but was content to give definitions in terms of ideas that's would be familiar to everybody." Then he goes onto the postulates of which the first is "a straight line may be drawn from any point to any other point." So then when he tells you what you can do with these things you can start to do math with it. I thought the mathematician all agreed this, but it's we common mortals tend to relapse and think of straight lines etc. as real things somehow out there about which we ought to be able to prove things. Oh I don't know.

    Anyway another thing. Euclid is also famous, unlike a modern day celeb, for nothing being known about him personally. So my question is - what is it assumed he was a man? Couldn't he have been a committee?
     
    Last edited: Dec 20, 2015
  20. Dec 20, 2015 #19
    Any problem with any use of language in Euclid should first be suspected to be a translation problem. The definitions I quoted are very old translations of vastly older texts, which are copies of copies of copies of copies of the originals.

    To me it's clear Euclid did not present points and lines as "things" at all, but as idealized concepts. There is no suggestion at all that we're going to find points, lines, planes, in nature. An authentic 30-60-90 degree triangle will only exist in the human mind. But, that's all the existence it needs to do math with it in the human mind.

    I am pretty sure all the experts agree that Elements was a compilation of many previously existing works on Geometry representing the combined wisdom on the subject of many schools. There's controversy about how much a Greek mathematician named Euclid actually contributed, but, for convenience sake, we now call it "Euclid's" Elements.
     
  21. Dec 20, 2015 #20
    Anyway, I want the OP to see this, because it might be the earliest know assertion a straight line is the shortest distance between two points, and since it's presented by Archimedes as axiomatic, I suspect all the Greek geometers took it the same way, as too obvious to require proof. So, the same is probably true about the shortest distance between a line and a point not on the line. :
     
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