Re: Why didn't ancient Greek Mathematicians use a string instead of a compass for their constructions?
- From: eaglesondouglas@xxxxxxxxx
- Date: Thu, 11 Dec 2008 07:41:10 -0800 (PST)
On Dec 9, 7:11 am, "Philippe 92" <nos...@xxxxxxxxxxxx> wrote:
eaglesondoug...@xxxxxxxxx wrote :
On Dec 9, 3:48 am, lbrt...@xxxxxxxxx wrote:
As you could tell from my previous question:
I am interested in the details of why have we Mathematicians used
certain means and ways both logical and mechanical to operate and
express ourselves
~
Using a string would have let them produce not only circumferences,
but also ellipses, so they wouldn't need conic sections
I'm not sure the study of conic sections by the ancient was motivated
by "extending the compass and straightedge".
Geometry was the issue.
...
It is for this reason geometry IS NOT mathematical.
Humm. I leave this opinion to YOU.
However a string (apart from the added ellipses(*)) allows to construct
a lot of things too. Of course straight lines (tightening the rope,
that's the way gardners and other house builders draw straight lines,
with a rope soaked with chalk). Of course a circle. So all what can
be drawn with compass and straight edge can be drawn with rope alone.
But as you said, it allows even more, as does just paper folding
for instance (allows trisection of angles, and construction of 7
sides regular polygons).
Some times ago, I played with this idea :http://mathafou.free.fr/pbm_en/pb220.html
However for the history of maths, I think nobody knows *exactly*
what was in the mind of ancien geometers.
I've been told that the discovery of irrationnality (sqrt(2)) gave
them a big shock !
Hence they tried to prevent the occurrence of such further shocks by
not allowing too powerfull tools, and limiting to just compass and
straightedge. The Idea is as is.
In that time, geometry was quite "contemplative", many proofs being
"just look" from a well drawn figure. Hence the search of exact
constructions in that time.
Then much effort has been made to make the geometry more rigourous.
First one being Euclides, then more recently Hilbert etc.
Then the geometry today is much more algebraic, considering sets and
metrics and manifolds etc. (but it IS mathematical)
Considering geometry constructions as just a game, we need however to
fix the rules. Changing the rules changes the results (what can and
can't be constructed).
As well known, the use of rope (the 12 knots rope) to construct
right angles was a current practice. Why didn't they choose the
'rope only' constructions ?
To mention also that even the ancient tried to overcome these limits,
by using specific curves (trisectrix, quadratrix etc) instead of just
circles and straight lines, also "Neusis constructions" like the
well known angle trisection by Archimedes (humm, not sure he was the
first) with a marked straightedge etc.
As there are mechanical tools to draw continuously conic sections,
you could use "construct by conic sections" instead of just compass
and straightedge (straight lines and circles being degenerate cases
of conic sections, we call that just "conic sections").
But these are quite "recent" discoveries.
A big problem also is that a straightedge is quite hard to make.
(How do you ensure it is perfectly straight ?)
But a compass is much more robust. Just suffice it is enough rigid.
Hence many geometers tried to discard the straightedge.
I mentioned Mascheroni in another post.
The study of inversion also gave a mean to *construct* a straight line
from scratch (from a circle).
The world of construction rules, and geometry fundaments is quite
wide...
(*) draw an ellipse with a rope. Proove that it is a true ellipse,
considering the diameter of the pencil etc...
The right method : use a LOOP of rope, going around 3 poles of same
non null diameter : two fixed poles, one moving (the pencil).
It is not if the 3 diameters are unequal.
Regards.
--
Philippe Ch., mail : chephip+n...@xxxxxxx
site :http://mathafou.free.fr/ ; (recreational mathematics)
The set as a rule becomes the issue. Angle location state defines
Geometry. A line as a set defines mathematics.
It is unhealthy to mix state. A function as a cause to set state
allows geometric functional sets though.
Just be reminded that functions are not just mathematical. Functions
preceed mathematical sets as a mapping of set element to ANY other
set.
You can call it mathematical geometry, but you just might call it
correctly as functional geometric state.
.
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