Subject: Re: [HM] Caratheodory Biography
From: Emili Bifet (bifet@attglobal.net)
Date: Mon Mar 20 2000 - 00:23:32 EST
[Antreas P. Hatzipolakis]
>
> 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?
On Sat, 18 Mar 2000, John Conway wrote:
>
> 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.
[deleted]
> 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. [deleted]
In the spirit of making these things better known to today's
mathematicians, let me mention that Caratheodory's Theorem would be,
for example, a direct consequence of The/ore\me II on page 43 of the
book "Introduction a\ la Ge/ome/trie Diffe/rentielle suivant la Me/thode
de H. Grassmann" by C. Burali-Forti published by Gauthier-Villars in
1897. This book, in turn follows Giuseppe Peano's own version of
Grassmann's ideas, in his book of 1888 "Calcolo geometrico secondo
l'Ausdehnungslehre di H. Grassmann." (This remarkable book deserves to
be better known, and read: it contains, for instance, a completely
modern account of vector spaces, including infinite dimensional
ones---putting the Goettingen Boys to shame. If Italian is a problem,
you may want to look at the English translation of the smaller 1896
"Saggio di calcolo geometrico" on pages 169-188 of "Selected Works of
Giuseppe Peano," ed. H.C. Kennedy, U. of Toronto Press, 1973. H.C.
Kennedy also included a translation of two chapters of the 1888 work,
but unfortunately he left out the far more important later parts.)
My reasons for mentioning Burali-Forti's book in particular are
two-fold:
1) it is fortunately available on-line at the Cornell Digital Library
http://moa.cit.cornell.edu/dienst-data/cdl-math-browse.html
(the following link should hopefully take you to page 43
note that applying Thm II [say to A - O, B - O, C - O, O] gives
Caratheodory's result.)
2) I believe that reading Burali-Forti's book provides a very good way
for a modern reader to learn these things. Such a reader will be
rewarded with new geometric insight into Exterior Algebra, Homology, de
Rham's Theorem, Currents, Geometric Measure Theory, ... Afterwards,
such a reader may wish to proceed backwards and read the first volume of
Grassmann's works (if the German is a problem, the book published by
Open Court in 1995 contains essentially a translation of this volume,
editorial notes included.) It may be a good idea to start with the
"Kurze Uebersicht ueber das Wesen der Ausdehnungslehre." Eventually,
reading Leibniz's letter of September 8th, 1679, to Huygens may bring
her back to the place where it all began. (The letter can be found, for
example, in pages 17-25 of G.W. Leibniz Mathematische Schriften II, ed.
C.I. Gerhardt, available on-line at
http://catalognum2.bnf.fr/html/i-frames.htm
Gallica=BnF=Bibliothe\que nationale de France. I would also love to
provide an on-line reference to Grassmann, but I am not aware of any. It
seems that poor Grassmann, once ostracized by his contemporaries, finds
himself nowadays ostracized by librarians.)
Best wishes,
Emili Bifet
PS I hope that what follows may help a modern reader trying to read
Burali-Forti or Peano:
E = Euclidean space (of dimension 3 say, but it could be an arbitrary
n;)
R = Real numbers;
E(1) = free real vector space generated by [the set] E = all real valued
functions on E having finite support;
E(2) = ... generated by [the set] E x E;
E(p) = ... generated by the product of p copies of E;
E(4) ---> R = linear map induced by oriented volume of tetrahedrons;
E(p) x E(q) ---> E(p+q) = obvious bilinear map induced by concatenation
= map given, in term of functions, by (f,g)(a,b)=f(a).g(b);
E(p) x E(q) ---> E(4) ---> R (whenever p+q=4) = bilinear map obtained
from previous one followed by oriented volume;
F(p) (p = 0, 1, 2, 3, or 4) (geometric forms of the p-th order) =
quotient of E(p) by the left kernel of the previous bilinear map ( i.e.
the kernel of the corresponding linear map E(p)--->E(q)* =~= R^{E x ...
[q times] ... x E} ;)
[free] vector (a particular form of the first order) = [equivalence
class of the] difference of two points, B - A;
F(p) x F(q) ---> F(p+q) (p+q <=4)([alternating] progressive product) =
bilinear map induced by the previous pairing;
segment (a particular form of the second order) = a product AB of two
points;
triangle (a particular form of the third order) = a product ABC of three
points;
bivector (a particular form of the second order) = a product of two
vectors;
trivector (a particular form of the third order) = a product of three
vectors;
et cetera.
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