Richard Heathfield
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Sorry to dump this on sci.math - just a matter of definition to clear
up. To save you ploughing through a lot of nonsense (below), I'll
summarise here. The question at issue is specifically regarding
Euclidean geometry. Euclid drew a distinction between "line" and
"straight line". Do modern Euclidean geometers do the same, or would
they nowadays consider that a line is straight by definition?
On 11/06/16 20:54, Peter T. Daniels wrote:
> On Saturday, June 11, 2016 at 3:11:36 PM UTC-4, Richard Heathfield wrote:
>> On 11/06/16 19:35, Peter T. Daniels wrote:
>>> On Saturday, June 11, 2016 at 11:09:14 AM UTC-4, Richard Heathfield wrote:
>>>> On 11/06/16 14:55, Peter T. Daniels wrote:
>
>>>>> I do not deny that after Book I I am lost in *Elements*, but I'm quite certain
>>>>> that in later books spheres are dealt with.
>>>> He did, yes. His definition is in Book XI: "When a semicircle with fixed
>>>> diameter is carried round and restored again to the same position from
>>>> which it began to be moved, the figure so comprehended is a sphere."
>>>> And he defines the semicircle in terms of the circle, and he calls the
>>>> bounding curve of the circle a "line", yes.
>>> Q.E.D.
>>
>> All you've demonstrated is that Euclid defined his terms in one way, and
>> nowadays we define them in another. This was not in dispute.
>
> No, I demonstrated that what you labeled as "In Euclidean geometry" was in
> fact in post-Euclidean geometry.
It may well be, whatever post-Euclidean geometry is. But my claim (that
a straight line cannot lie flat on the surface of a sphere) is also true
in Euclidean geometry (but the claim is /not/ true in elliptic geometry).
>
>>>> And nowadays we would
>>>> consider that to be outdated.
>>> If you had introduced your comment with "In post-Euclidean" or "non-Euclidean
>>> geometry," there would have been nothing to complain about.
>>
>> There is still nothing to complain about. Euclidean geometry continues
>> to be known as Euclidean geometry, even though we have moved on somewhat
>> since Euclid's day. For example, we don't insist that Euclidean
>> geometers discuss Euclidean geometry in Ancient Greek. Nor do we insist
>> on their being nailed to Euclid's precise wording. We insist only that
>> Euclid's five postulates hold. To label such a geometry as
>> 'non-Euclidean' would have been pointlessly confusing to most people.
>
> The definition of "line" as 'straight line' is simply NOT Euclidean.
Enough already. Cross-posted to sci.math - let the mathematicians have
their say.
>
>>>> We can forgive Euclid for being outdated
>>>> occasionally, given that he lived well over two thousand years ago, but
>>>> that doesn't mean we should continue to be outdated nowadays. If I come
>>>> across a 19th century treatise on number theory that considers 1 to be
>>>> prime, I don't say it's wrong - because at the time it was considered
>>>> correct. But the same usage in the 21st century would be a mistake,
>>>> because we have redefined primality to exclude 1.
>>>
>>> But you don't say, "In the 19th century 1 was not considered a prime number."
>>
>> That's right. I don't. But we're chasing a wild goose. All we're talking
>> about here is the definition of "line" (vs "straight line"), and Robert
>> Bannister specifically talked about a "straight line".
>
> No, all we're talking about is what you mistakenly claimed Euclid said.
You are misremembering. I have not misquoted Euclid anywhere.
<snip>
>>>> I'm not sure what point you're trying to prove here. Let's look at the
>>>> claim under dispute: it was claimed (and I'm rephrasing here but, I
>>>> hope, retaining the intended meaning) that, when an angle of 180 degrees
>>>> is subtended, the two 'ends' of the angle lie in a straight line.
>>>
>>> Actually no, the original claim was that from North Pole to South Pole is 180 deg.
>>
>> I am not referring to the original claim. I am referring to the claim
>> under dispute. That is why I said "the claim under dispute". The claim
>> under dispute is the claim that we have been disputing, which is a
>> rather pointless one about Euclid's prototype definitions.
>
> It is about your mischaracterization of them.
Well, it's a dispute, so you're bound to stick your oar in, and I can
hardly fault you for that, but obviously I disagree.
<snip>
>> And, in so doing, he provided an
>> opportunity for an interesting discussion about geometry.
>
> There was nothing interesting about your mistake.
Well, I don't agree that I made a mistake, and neither do I agree that
the discussion hasn't been interesting.
>>>> In elliptic geometry, the term "straight line" is used to describe what
>>>> we would ordinarily call a great circle. Let us imagine that such a
>>>> straight line has been drawn. If you mark a point on that line, put the
>>>> cross of a protractor on the point, and measure the angle subtended by
>>>> the two drawn line segments, you will find that it is 180 degrees.
>>>
>>> On the surface of the earth, a "straight line" journey necessarily follows a
>>> portion of a "Great Circle." Since the original observation was couched in terms
>>> of North Pole and South Pole, that's the relevant frame of reference.
>>
>> I would agree. So the question becomes: if we define two points A and B
>> as lying on the same great circle route drawn on the Earth, and we draw
>> a line segment from A to a new point C (also on the surface of the
>> Earth) such that the angle between AC and AB is 180 degrees, do AB and
>> AC lie on the same great circle route? And the answer is that they do.
>>
>> And quod, Mr Daniels, is what really erat demonstrandum.
>
> There was no dispute about that (since you'd omitted it from consideration).
You think I omitted from consideration the very point that I discussed
in my first reply on the subject? Okaaaaaay....
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
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