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\centerline{\bf Review of ``Poincar\'e and the Three Body
  Problem'' by June Barrow-Green}

\centerline{\bf Daniel Henry Gottlieb}
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In a work of impressive scholarship, 
the author takes us through
the history of the $n$ body problem from Newton to 
the present.  The center of her story is the prize competition in
honor of the 60th birthday of King Oscar II of Sweden in 1889.
With royal patronage, with the most prestigious mathematicians as
judges, and with the most momentous  
mathematical problem of Civilization as a topic, it had captured
the attention of the mathematical world. And the winner was ...
Henri Poincar\'e ...  with a manuscript that 
had a major error!

The paper was due to be published on the King's birthday a few
weeks hence, when Poincar\'e himself
discovered the false result. The difficulty of his position was
enormous. 
An error in a paper so highly honored not only would be
a great personal embarrassment, but it would damage the reputations
of the judges and the organizers of the competition as well as
ruin the Kings birthday.

Poincar\'e wrote the letter (surely the most difficult one a
mathematician ever had to write), stopped the presses, paid for
the printing costs (which exceeded the prize money by 1000
Kroner), and worked feverishly on a new manuscript, 
which was printed a year later.
The letter and the first suppressed manuscript remained hidden
in the Mittag-Leffler Institut archives and were only recently
rediscovered.

The author analyzes the suppressed flawed manuscript along with
the published corrected copy. 
What underlay Poincar\'e's error ``is arguably the first
description of chaotic motion within a dynamical system.'' The
author goes into mathematical detail in tracing the influence of
this manuscript, and of later ones 
by Poincar\'e
on the subjects of
Differential Equations, Dynamical Systems, and Celestial Mechanics. 
This 
mathematical detail
should be interesting for the practitioners of those
disciplines, for she has a 
clear way of describing research.
For the rest of us, she tells  
about the controversy between the dynamical astronomers and
Poincar\'e, the final solution to the three body problem, the
mathematical personalities and politics of the competition, and
much else.  Her scholarship gives  
a firm historical base
for reflections about what a mathematician really is.

For example, Arthur Jaffe and Frank Quinn's controversial 
article [1] 
discusses the issue of published results with inadequate proofs.
Among the cautionary tales  mentioned is
Poincar\'e's discovery of homology. ``Poincar\'e claimed too
much, proved too little, and his {\it reckless} methods could not
be imitated.  The result was a dead area which had to be sorted
out before it could take off.''  The context of these remarks
imputes a kind of dishonesty to Poincar\'e, and claims it
retarded the subject for years. 
However, in view of Poincar\'e's letter, we can ask the
author of those lines 
whether it seems to him now that Poincar\'e was dishonest, and we
can inquire if he himself would write such a letter if he were in
Poincar\'e's position. 
As far as the ``damage'' done by Poincar\'e, 
I point out that 
some of the greatest
mathematicians of the time took fifty years 
before they finally got
homology right, and in the process they fundamentally changed the
way we view almost all of Mathematics.

Practitioners of mathematics follow two
historical traditions.
One stems from the dawn of Civilization, the
other arose in the time of the Greeks. In the older tradition,
Mathematics is the handmaiden of the Arts, Science, and Industry.
In the Greek tradition, Mathematics is the Queen of Knowledge,
the only real way to Understand Nature.

By ``Understand Nature'', I mean understand it in the way 
that a Mathematician understands Mathematics:
Clearly, distinctly, without ambiguity.  
To paraphrase Galileo:
Once one tastes this kind of Knowledge, he can never be satisfied
with a less perfect kind.
Only a Mathematician can taste this kind of Knowledge.
(Here I mean Mathematician in an inclusive sense,
as opposed to merely a
member of a particular profession.) 
This kind of Knowledge makes 
Mathematics the Queen of the Sciences, and she will reign
forever.

But Mathematics is also the Handmaiden of the Sciences. It is a
collection of tools to solve problems, to obtain answers, 
to describe and to measure and to name. You use it to build a
bridge, to  survey the land, or to navigate the sea.  
It perfected the masterpieces of our great painters and cast the
horoscopes of our superstitious ancestors.  This tradition is
much older than the Greek tradition of Mathematics as a pure kind
of knowledge, and for most well educated people it forms their
view of what Mathematics is. Those who hold to this tradition
may practice their mathematics with skill, 
but the mathematics is secondary to other considerations.
I will call these people Practitioners.

Now among the fascinating things in this story is that a
Practitioner named Hugo Gyld\'en, upon obtaining information
about Poincar\'e's original prize winning paper, 
claimed he in fact
had already published all of Poincar\'e's results.
This led to a long controversy between those Practitioners
called Dynamical Astronomers, and the Mathematicians. 
The problem was 
that the Mathematicians and the Astronomers had
different ideas about what convergence of a series means:

``To illustrate how mathematicians
and astronomers differed over this question, Poincar\'e compared the possible
interpretations of the following two series

$$\text {$\Sigma (1000)^n/n!$ \text{   and    }  $\Sigma n!/(1000)^n$} $$.

He argued that a mathematician would consider the first convergent and the
second divergent, while an astronomer would label them the other way round."

This must be the best of the math versus
physics jokes, because it is true! Yet Poincar\'e did not convey
any criticism. He merely wanted 
to explain the difference
to eliminate misunderstandings. 
He understood that truncating a divergent
series whose initial terms decrease fast could produce numbers
that are useful in practical problems, but he pointed out that
such methods should not be used to prove theoretical results.
And he observed that for practical computations it really does
not matter whether or not the series converges:
What is important is to have some idea of the upper bound of the
errors involved.

Poincar\'e always said he 
% ``said that he''
learned a great deal from these Practitioners, including
Gyld\'en. Most of us would react in the spirit of Hermite, who
`` was not impressed 
by Gyld\'en's grasp of analysis, describing Gyld\'en as a ghost
from a bygone age, who had been left behind as the world of
analysis transformed about him.'' 
It turns out, though, that Poincar\'e had the right
point of view, because in 1909, the Finnish astronomer Karl
Sundman completely solved the three-body problem!
Given an initial position, he could produce a convergent series
giving
the positions of the bodies for all times.

Wait a minute, I didn't know that
the Three Body Problem was solved.  I'll bet you didn't either.
`` Sundman's work 
seems to have been almost forgotten. Why did such an important
and long awaited work almost fade into obscurity?''  Think about
it!

The $n$ body problem can be thought of as the most fateful
problem in all of Mathematics. One might say that the
mathematician Galileo Galilei ``solved'' the one body problem by
assuming as an axiom
that a body moves with uniform motion in a straight
line. Reasoning with other axioms, he showed that a projectile
follows a parabolic path. To objections that no one has seen a
body move forever in a straight line, he would say: Let us derive
the mathematics, and compare the results to those we actually
observe. If their differences can be explained by other effects,
then the axiom is reasonable. If there is a disparity, then the
axiom should be discarded. Thus Galileo followed in the tradition
of Archimedes and used Mathematics to Understand Nature.

The two body problem was essentially solved by Newton in 1687.
Newton laid down his three Laws, the first two adapted from Galileo, 
plus the fourth Law of Gravity.
With these axioms mathematicians could understand the workings of
the solar system, and they strove
to develop methods of calculating. 
This stupendous achievement, Newtonian Mechanics,
led to an entire reorientation of Western Culture. The following century, 
called the Age of Reason, 
found Mathematical ideas applied in every field as 
people
tried to emulate Newton's clarity.

How could the solution of the three body problem fade into
obscurity? Well, 
the first reason is that Sundman's series does not
converge according to the Practitioners.  
The second reason is that it is an algorithm
conveying little insight: although it is precise, it does not
add much 
to Understanding Nature. 
The third reason 
is that the Physicist Albert Einstein, noting some slight
inadequacy in Galileo's solution of the one body problem,
propounded a new solution, and thereby became a Mathematician.

I thoroughly enjoyed June Barrow-Green's book. I have written
things here I would not have dreamt of saying before reading it.
For me, the center of the work is Poincar\'e's letter; 
for now we can show the Practitioners, and the World,
just what we Mathematicians are.

REFERENCE

[1] Jaffe, Arthur; Quinn, Frank "Theoretical mathematics": toward a
cultural synthesis of mathematics and theoretical physics. Bull. Amer. Math. Soc. (N.S.) 29
(1993), no. 1, 1--13. 
\bye
