Who and when - simple piece of geometry

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  • #1
julian
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Main Question or Discussion Point

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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  • #2
DaveC426913
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That's part of Euclidean metric. I'm not sure he first proved it though.
 
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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?
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."
 
  • #4
julian
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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."
I'm only asking about "The shortest distance from a point to a LINE".
 
  • #5
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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.
 
  • #6
DaveC426913
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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:
The primal way to specify a line L is by giving two distinct points, P0 and P1, on it. In fact, this defines a finite line segment S going from P0 to P1 which are the endpoints of S. This is how the Greeks understood straight lines, and it coincides with our natural intuition for the most direct and shortest path between the two endpoints.
which characterizes this idea as "intuitive," something that suggests the greeks felt no urgency to construct a proof.
 
  • #8
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I'm only asking about "The shortest distance from a point to a LINE".

Yes I know. I was trying to show something amusing, not insult you. Sorry.
 
  • #9
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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.
 
  • #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
 
  • #11
DaveC426913
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...this cannot be a question about lines ... it's about distances...
Well, yes. Of the four posters to this thread, each has mentioned distances explicitly.

I think it's about the number of times one segment of a line fits into another of the same or a different line.
Not sure I follow.
 
  • #12
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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...
Euclid's Elements said:
Definition 2.
A line is breadthless length
Definition 3.
The ends of a line are points.
Definition 4.
A straight line is a line which lies evenly with the points on itself.
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.
 
  • #13
epenguin
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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.
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?

Not sure I follow.
What makes you think you're meant to? It Can Be Shown I tell you! :oldbiggrin:
 
  • #14
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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?
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.
 
  • #15
epenguin
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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.
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.
 
  • #16
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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.
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.
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.
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:

Archimedes said:
1. "Of all lines which have the same extremities the straight line is the least."
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.
 
  • #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."
 
  • #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?
 
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  • #19
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In answer to the zoo be, 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.
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.

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.
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.

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?
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.
 
  • #20
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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. :
Archimedes said:
1. "Of all lines which have the same extremities the straight line is the least."
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.
 
  • #21
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I'm flipping through my copy of Euclid's Elements right now and I don't think I can find a proof of that exact statement, but I think it could easily be taken as an alternate meaning of Proposition 13 or found easily from Proposition 48, both from Book One:

Proposition 13 http://aleph0.clarku.edu/~djoyce/elements/bookI/propI13.html
Proposition 48 http://aleph0.clarku.edu/~djoyce/elements/bookI/propI48.html

That the shortest distance between two points is a straight line is one of Euclid's starting assumptions ("axioms"), hence why spaces where this is true are called "Euclidean".
 
  • #22
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I'm flipping through my copy of Euclid's Elements right now and I don't think I can find a proof of that exact statement, but I think it could easily be taken as an alternate meaning of Proposition 13 or found easily from Proposition 48, both from Book One:

Proposition 13 http://aleph0.clarku.edu/~djoyce/elements/bookI/propI13.html
Proposition 48 http://aleph0.clarku.edu/~djoyce/elements/bookI/propI48.html

That the shortest distance between two points is a straight line is one of Euclid's starting assumptions ("axioms"), hence why spaces where this is true are called "Euclidean".
No, that is true of many spaces. A space is Euclidean if and only if it has the metric, a^2+b^2+... = r^2.
 
  • #23
julian
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Thanks for all your responses. Sorry not replying until now but I had an eye injury that meant I couldn't read anything for a while (and then it was x-mass). I've been going through people's thoughts and what people have found out now. Thank's for that. Still digesting it all.

A preliminary to my original question would be something zoobyshoe said

"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."

Agreed that he would first have to establish this.

zoobyshoe then later mentioned

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.
And

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. :
The statement "...the shortest distance between a line and a point not on the line" - could possibly be too obvious to prove - agreed.
 
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  • #24
julian
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The statement would have been obvious to prove...

So you have a point C and a line AB. You want the shortest distance from the point C to the line AB. Proposition 12 and the diagram given there opens up an obvious proof. Here I will use the notation used in Proposition 12:

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

In this proposition he has: "I say that CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it."

Utilizing the construction given in this proposition, you could then say that as the point D can be chosen arbitrarily, any point on the line AB can be identified with either E or G by appropriate choice of point D.

It is then intuitively obvious that the line segment CE (or CG) is longer than the perpendicular line segment CH. However, we can be more rigorous about this by

(i) simply noting that CG is equal to CF, that CH is less than CF, and as such CH is less than CG.

(ii) or using Pythagoras's theorem to prove CE (or CG) is longer than the perpendicular line segment CH. Pythagorous (c.570 - c. 495 BC).

Hence CH is the shortest distance from the point C to the line AB.
 
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