Subject: Re: [HM] Caratheodory Biography
From: John Conway (conway@math.Princeton.EDU)
Date: Sat Mar 18 2000 - 10:35:38 EST
On Fri, 17 Mar 2000, Antreas P. Hatzipolakis wrote:
> The following theorem, due to Caratheodory, is known as "Caratheodory
> Relation": A triangle ABC is given. Consider a point O inside the triangle.
> Then:
> --> --> -->
> E_A * OA + E_B * OB + E_C * OC = 0,
>
> where E_A, E_B, E_C are the areas of the triangles BOC, COB, and AOB resp.
*************************** Note: middle one should be COA - JHC
> Is it in fact a Caratheodory theorem, or a misattribution?
I wouldn't exactly call it a misattribution, but I'm sure the
great Caratheodory himself wouldn't claim it as his theorem, because
an equivalent result (in much greater generality) is in Grassman's
Ausdehnungslehre, and I think very probably in Mobius' little book
on the barycentric calculus before that.
Let me explain the context. The main idea of the barycentric
calculus is the use of barycentric coordinates, defined in this
dimension by saying that (X:Y:Z) are the coordinates of P just
if P is the center of gravity of masses X,Y,Z placed at the
vertices A,B,C of the triangle of reference. Now Mobius points
out that one can take these masses to be the areas BOC,COA,AOB (so
obtaining the particular normalization later called "areal" coordinates).
Now the result X.OA + Y.OB + Z.OC = 0 does appear in Grassman,
along with many generalizations, as part of his theory of exterior
algebra, and since the equivalence of the barycentric and areal definitions
was common knowledge, I think we can safely say that Caratheodory
probably regarded this problem as a geometrically transparent version
of a theorem that was well known to most professional mathematicians
of his day.
Unfortunately, these things are less well known to today's mathematicians
(except for Hyacinthos members!!), which I think accounts well enough
for the name. I am reminded of a story told by Klein, according to
which the teaching of calculus was at one time forbidden in German high
schools, because it was regarded by someone in the Education Ministry
as not rigorous. This meant that textbook authors were forced to avoid the
notation and terminology of the calculus, but they managed to get past
this by presenting the standard arguments in a geometrical version that
wasn't identifiable to the Ministry officials. As a result, the beginning
theorems of the calculus were all attributed to the authors of these
textbooks for quite some time!
John Conway
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