In the comments to

*Rectangular Circles - Yet Another Responseto Mathematician*, I wrote what I thought (with some hubris) was a great argument, a killer argument:
I get the point that you are
making here (or at least I

When you say (paraphrased) "there could be far more chords with c in [R/2,R] than in [0,R/2]" you are, I presume, assuming a "proper" mathematical circle/disc - we are talking about Euclidean space and not talking about curved space, or anything tricky like that. If so, I'd have to ask, on what basis, other than your selection process for problems like this, can you suggest that we might have more possible chords (and thus more possible lines defined by extending those chords out to infinity) passing through the interval [R/2,R] than any other interval of the same length? The circle/disc under consideration is essentially undefined as far as location, size and rotation go, so we should (reasonably) be able to change the locus and not have our answer change on us - but what you are suggesting is that if we shift the locus up by R, and rotate the circle/disc by π, then we'll change the number of lines passing through the intervals [0,R/2] and [R/2,R]. Ditto if we expand our circle/disc by a factor of 2 while retaining the locus at the notional (0,0).

We could even have two overlapping circles/discs, both of radius R, one with a locus at (0,0), the other with a locus at (0,R). This would mean that you'd simultaneously have more lines passing through [R/2,R] (as defined by chords in the first circle/disc)

This seems odd to me. Does it not seem odd to you?

**think**I do). It's why I've talked about the "set of ALL chords".When you say (paraphrased) "there could be far more chords with c in [R/2,R] than in [0,R/2]" you are, I presume, assuming a "proper" mathematical circle/disc - we are talking about Euclidean space and not talking about curved space, or anything tricky like that. If so, I'd have to ask, on what basis, other than your selection process for problems like this, can you suggest that we might have more possible chords (and thus more possible lines defined by extending those chords out to infinity) passing through the interval [R/2,R] than any other interval of the same length? The circle/disc under consideration is essentially undefined as far as location, size and rotation go, so we should (reasonably) be able to change the locus and not have our answer change on us - but what you are suggesting is that if we shift the locus up by R, and rotate the circle/disc by π, then we'll change the number of lines passing through the intervals [0,R/2] and [R/2,R]. Ditto if we expand our circle/disc by a factor of 2 while retaining the locus at the notional (0,0).

We could even have two overlapping circles/discs, both of radius R, one with a locus at (0,0), the other with a locus at (0,R). This would mean that you'd simultaneously have more lines passing through [R/2,R] (as defined by chords in the first circle/disc)

**more lines passing through [0,R/2] (as defined by chords in the second).***and*This seems odd to me. Does it not seem odd to you?

Mathematician
replied (with my current responses interspersed):

> "set of ALL
chords"

Wow, maybe I understand what you mean, but it would be odd. We have a given circle, right? When you say the "set of ALL chords", are you including the chords that are NOT inside the given circle (but inside another circle somewhere else ...)?

Wow, maybe I understand what you mean, but it would be odd. We have a given circle, right? When you say the "set of ALL chords", are you including the chords that are NOT inside the given circle (but inside another circle somewhere else ...)?

Was that your point all along for repeating "ALL chords" all the time? It would make sense with the rest of the argument:

I first talked about
"ALL chords" in a response to a comment on

*Triangular Circles*. In*Mea Culpa*, I put some effort in to explain what I meant by "ALL chords" – there is hopefully no indication whatsoever that I had any thought about considering all chords in all circles, and thus including chords that are not in the circle being considered. No, I meant "ALL chords"**.***in the circle being considered*
My apologies for
not making that sufficiently clear.

Sadly, the banks
rarely have this sort of confusion, so when I go and tell them that I want to
take out "all the money", they don’t pop out the back and give me every
single dollar from everyone’s account … they just give me the money that that
was in my account.

> on what basis can you
suggest that we might have more possible chords (and thus more possible lines
defined by extending those chords out to infinity) passing through the interval
[R/2,R] than any other interval of the same length?

Since the beginning, you are thinking of chords as the intersection of a straight line with the disc. That's a great characterization and a good way to get chords. (but not the only one, as we both know)

Since the beginning, you are thinking of chords as the intersection of a straight line with the disc. That's a great characterization and a good way to get chords. (but not the only one, as we both know)

So if I'm not mistaken, for your point of view, there is an existing set of all straight lines on the entire plane (like an infinite net), and you are just taking the intersection of this existing set of straight lines with a given disc. And you say that if you move the disc around, it will not cross the same straight lines, but the answer to Bertrand question should remain the same. Am I correct to assume that this is more or less your reasoning?

With this interpretation you are absolutely correct and agree with Jaynes argument. This is a mathematically correct argument.

At least I have
that right!

But that's not the only natural
point of view on this problem.

See, I'm taking another characterization for chords. For me a chord is a segment between two points on the circle. So there is nothing "outside" my circle. I have no reason to extend a chord out to infinity. The chords are not intersection of lines with the disc, they are segments inside the disc! There is no reason to consider objects (lines) that are not chords on the given circle, don't you think?

See, I'm taking another characterization for chords. For me a chord is a segment between two points on the circle. So there is nothing "outside" my circle. I have no reason to extend a chord out to infinity. The chords are not intersection of lines with the disc, they are segments inside the disc! There is no reason to consider objects (lines) that are not chords on the given circle, don't you think?

So, If I change the locus of the circle in the plane, the chords are moving with it. If I double the size of my circle, then the chords inside it will double their size. If I rotate the circle, the chords will rotate with it. So the final answer to Bertrand question will not change at all.

And with that point of view, it's perfectly natural to have more chords close to the rim than close to the locus. You only think it's odd because you are thinking of an existing "net" of straight lines on the plane, and you place your circle on that existing net of lines. But from my point of view, there is no "net" of existing straight lines.

I note that you
write here "The chords are not (the) intersection of lines with the disc". In one of your comments at

*Triangular Circles*, you wrote (my emphasis):
A circle is a
1-dimensional curve in the plane. A disc is the 2-dimensional surface that is
enclosed by the circle. So it's important to make the distinction, whether you
talk about the endpoints (which are on the circle) or the midpoint (which is in
the disc) of

**.***a chord (which is the intersection of a line and a disc)*
Perhaps you can see
why someone might get confused.

So my problem with
this approach, and I’ve hinted pretty strongly at it (that is by writing it),
is that as you have said yourself a chord

**the intersection of a line and a disc – irrespective of how you select that chord – and a chord is thus also a segment of a line – again irrespective of how you select the chord (it just happens to be the segment of the line that intersects with the disc).***is*
If you

**have a circle and nothing else, then I guess I would have to agree with you, at least on the basis of my ignorance with respect to the implications and also in recognition of your authority as a Doctor of Mathematics. However, immediately after invoking this free-floating circle you refer to something***could***the circle – specifically when talking about changing the locus, altering the size and rotating it. You have thus invoked an external reference plane on which the circle/disc rests. Would you not agree that it is meaningless to talk about translational, rotational and scalar invariance if the circle is all there is?***outside*
It seems,
therefore, to me, that you are attempting to have your cake and eat it too.

Perhaps there was
something in Bertrand's original phrasing that leads you to think that we can
talk about a free floating circle, rather than one embedded in a plane. I just don't know. Unfortunately, my French is that of a rather
forgetful schoolboy, I remember bits and pieces, I get the general gist of a
menu or a wine list, but I never got to the stage at which I might have
deciphered

*Bertrand's original text*. If only there were some French speaking mathematician I could call on to interpret …*This paper*, interesting, explicitly refers to a plane, or rather

**plane: "Consider a disk on the plane with an inscribed equilateral triangle." In the afterword, the author writes (rather pleasingly from my point of view):**

*the*
In his pointedly titled paper

*The Well-Posed Problem*, (Jaynes) applies this principle to the Paradox of the Chord with success, uniquely identifying the uniform distribution over the distance between the midpoint of the chord and the center of the disk as the correct choice of measure, which he then proceeds to verify experimentally.
The use of the term
"correct" is particularly satisfying, although I'm sure that there's
some reason why my understanding of the term "correct" is limited and
that that lack of understanding will be swiftly addressed.

> We could even have two
overlapping circles/discs, both of radius R, one with a locus at (0,0), the
other with a locus at (0,R). This would mean that you'd simultaneously have
more lines passing through [R/2,R] (as defined by chords in the first
circle/disc) and more lines passing through [0,R/2] (as defined by chords in
the second).

If I understand correctly, with your point of view, if you have two circles in the position you gave, there is as much chords in the interval [0,R] than in the interval [R,2R], right? With your point of view, the fact that we have one, two or seven circles, do not change the "density" of chords at all, is that correct? This seems odd to me.

If I understand correctly, with your point of view, if you have two circles in the position you gave, there is as much chords in the interval [0,R] than in the interval [R,2R], right? With your point of view, the fact that we have one, two or seven circles, do not change the "density" of chords at all, is that correct? This seems odd to me.

With my point of view, each circle has its own set of chords, so the "density" of chords will be higher in the intersection of both disc. So there will be "twice as much" chords in the interval [0,R] than in the interval [R, 2R], because there are two set of distinct chords.

If we permit a
single mathematical line to "carry" multiple, overlapping, distinct chords,
then I don’t see a major problem with the density of chords changing as you
overlay circles (ie you could conceivably have ten identical circles on top of
each other, with an infinite number of chords, each of which is replicated ten
times). But in the context that you
quoted, I was not talking of chords per se, nor was I really thinking about the
chords in multiple circles (except obliquely).
Perhaps I should not have even mentioned overlapping circles at all,
because this has only served to confuse.

You seemed to
understand what I was saying earlier, when you talked of a "net of
straight lines on the plane" (my recollection is that all lines are
straight, but perhaps you were just clarifying this for my benefit).

There will be no
line that passes through the circle (across the disc) which is not also
coincident with a chord, right? (By this
I mean that there will be a segment of the line which directly corresponds with
a chord, the ends of which lie on the circumference of the circle/disc - the
segment shares the same length, gradient and endpoints as the chord such that
one could almost consider them to have the same identity. When I said "carry" above, I mean
to imply that such a chord is, in a sense, lying on the line with which it
shares a segment. It is possible, with
multiple circles, for there to be multiple segments of the line, perhaps
overlapping, lying on top of each other … perhaps entirely identical

**overlapping. This might not be standard phrasing, but I hope that you can understand my intention.)***and*
There is an
infinite array of lines that pass through the circle, and a(n infinite) subset
of those will intersect with a notional y-axis.
With your preferred method of selecting (or identifying) chords, via
their endpoints, it seems to me (and even to you, apparently) that there will
be fewer

**that intersect the y-axis at the locus of the circle than at the circumference – this is what does not appear to be justified. Spiriting the circle out of this universe does not appear to be justified either, but perhaps the problem as originally phrased does demand it and I am simply unaware of that aspect of it.***lines*
---

Would it help to
move away from circles for a moment? I
understand that there are certain things about circles that might lead you to
favour endpoint generation of chords.

I was thinking of a
similar problem, but involving a square (squared circles!) What is the probability that, on selecting at
random an s-chord (my term for the equivalent of a chord within a square) the
endpoints of which do not share the same side of a square, the selected s-chord
is greater than the length of the longer sides of an isosceles triangle which
has one of the sides of the square as its base (√5L/2
where L is the length of the sides of the square)?

This does, to me,
seem a more complex question to answer.
I'm tempted to say that the endpoint approach will give, once again, a
result of 1/3. We don't seem have an
equivalent of the other two approaches, because s-chords are not as constrained
as chords - but perhaps there is a way of thinking about them using one side of
the square as a reference.

But that's not what
I was thinking about specifically. What
I was thinking about is the density of s-chords. Would you expect to see them clustered around
the edges of the square (or sides), or at the corners (or vertices), and more
sparsely represented in the middle of the square?

---

Why did I call your
argument "circular"? I was going
to weave that into my response above, but there was no obvious or natural place
for it. So my explanation will have to
stand on its own.

My argument is that
a circle/disc doesn't exist outside of the plane on which it rests. This in turn means that the "net"
of mathematical lines of which the plane (conceptually) consists cannot be
naturally separated from the disc, and hence from the chords (each of which is
the intersection of a line and a disc).
If you ignore this, and consider chords as merely what you get if you
connect two points on the circumference of a circle, a circle which (in my
view) is strangely divorced from mathematical reality, then sure, you can float
this circle around, squeeze or stretch it and spin it around. With such an independent circle there is no
need for invariance of any kind (translational, scalar or rotational) - because
the circle is the circle is the circle.
If this is not circular …

(Please do note
that this is, at least in part, a joke - a poor excuse for a pun. My more serious efforts lie above this
section.)

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