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\noindent{\bf K\'aroly JORDAN}\\
b. 16 December 1871 - d. 24 December 1959
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\noindent
{\bf Summary.} K.Jordan, best known for his work on probability theory and finite
differences, was a founder of the Hungarian School of Probability.
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K\'aroly Jordan's path to mathematics, and to probability and
statistics in particular, was not straightforward. Coming
from a well-to-do family, he had the luxury of studying a profession
practically wherever he wanted to, and when his interest 
changed he could go back to study, or to train himself in, a new
field. Indeed, after having studied chemistry at \'Ecole Pr\'eparatoire
Monge in Paris and at the \'Ecole Polytechnique in Zurich, and become
a chemical engineer in 1893, he accepted an appointment
at the University of Geneva where he concurrently studied physics and
obtained the degree of Docteur \'es Sciences Physiques. 
He then gained the position of Privat Dozent in physical
chemistry at Geneva.

When Jordan returned to his birthplace, Budapest, Hungary, in
1899, and could not find a suitable job as a chemical engineer, he
entered P\'azm\'any P\'eter University of Science (the present
L. E\"otv\"os University of Budapest) and studied astronomy and
seismology. He ultimately became the director of the Seismological
Institute of Budapest which position he held up to 1913. During this
period his interest turned increasingly 
towards probability theory and to mathematics in general. This
was due to his interest in
understanding and scientifically explaining seismological and
meteorological data. His mathematical interest was very
broadly based, and he distinguished himself in subsequent years not only
in probability and statistics but also in the theory of differences and in
approximation theory.

The First World War appeared to interrupt Jordan's free movement
between subject matters and positions and he was appointed a teacher
of mathematics, physics and meteorology at a Military School in
V\'arpalota (close to the Bakony mountains of Hungary).
However, it was this teaching experience, combined with his earlier
publications on data from meteorology and seismology, which
determined the direction of all his subsequent work. 
After the war he was invited to lecture on statistics at
P\'azm\'any P\'eter University, from which position he quickly
switched in 1920 to the University of Economics of Budapest, where he
went through the ranks, culminating in his Professorship in 1933. His
lectures now extended to probability theory, statistics and the theory
of finite differences. He remained at the University of Economics
until 1949, a year in which the university was reconstructed into a
more structured and politically oriented institution. Jordan just
moved on, taking his free-style and very popular lectures to the
Technical University of Budapest.

Jordan did pioneering work in Budapest. There was no country
other than Hungary in continental Europe where there was 
systematic teaching of probability and statistics as early as
1920. His lectures always contained new results in probability theory
and applied statistics. 
They inspired a
great many people to devote their own lives to probability and
statistics: a school was born.

Jordan
was born in Budapest on December 16, 1871, and received his primary
and secondary education there. Then, during the ten year period 
1889 to 1899 he studied and worked abroad as an engineer. Only after
his return to Hungary in 1899 did he become a statistician and a
mathematician. Apart from his extensive travels to conferences, he
remained in Hungary all the rest of his life. He
married Marie Blumauer during his stay in Geneva in 1895 and she
died at the birth of their third child in
1899. Back in Budapest, in 1900 he married Marthe Lavall\'ee and this 
marriage also produced three children. The large family was a
great happiness to him and they remained very close
even when the children had grown up. 

Jordan frequently published in
Hungarian; he was particularly keen to contribute to mathematical
journals read by students.
Two of his
notable books also had their first publication in Hungarian: one on
mathematical statistics in 1927 and one on classical probability
theory in 1956.  The
publication of the probability book was a bitter-sweet experience
for Jordan: since it contained his life-long contributions to
the foundations of probability theory and
statistics, he was glad that it was finally out in 1956, but since he had
completed the manuscript in 1946 and submitted it for publication in that
same year, the long period of waiting and the associated
correspondence on the manuscript made him somewhat bitter.
During that period the publication of a manuscript was not
judged merely on its merits, but a very few people decided who
published what and when. Since Jordan wanted the book to appear in
Hungary, he chose to wait rather than translate it and publish it
elsewhere. 

The year 1956 was a memorable one for Jordan: he had hardly put
a copy of his new book into his magnificent library when a tank
fired into his house and the whole library, together
with all his possessions, burned down. This happened during the short-lived 
Hungarian uprising. The loss of the irreplaceable Jordan library was 
a serious blow. One of his 
hobbies had been to collect original works of mathematicians of
historical note, and he had some from the
15th century. Out of his 5000 volumes, about 1000 were rare books. By an
agreement between Jordan and the Hungarian Academy of
Sciences, the library would have become the property of the Academy
after his death.

Another hobby of Jordan was to travel and to explore nature. 
He participated in
the determination of the depth of the lakes Balaton and H\'eviz, and
reached several peaks of the T\'atra mountain some of which, as one of
his pupils, the distinguished probabilist, Lajos Tak\'acs
discovered, actually bear his name. He set and kept the speed record
by a sailboat for several years on lake Balaton,
When he was young, he travelled all over Europe on a bicycle,
but he switched to motorcycle
when he reached the age of 60. He died in Budapest on
December 24, 1959 at the age of 88.

Jordan's work was widely
recognised both in Hungary and abroad and he received
many honors. He was awarded the
K\"onig Prize in 1928 
by the E\"otv\"os Mathematical and Physical Society of
Budapest. He was elected a corresponding member of the Hungarian
Academy of Science in 1947 and received the Government's Kossuth Prize
in 1956. 

The work of Jordan and his contemporaries
was done without the benefit
of a systematic theory of probability as we know it today. But Jordan
and others did not feel that such a theory was lacking. They gave a
precise definition of probability and argued using 
this. Jordan's concept was quite broadly based, and in some
respects it comes close to current ideas. This allowed
Jordan to be more philosophical about the basis of probability and
the theory that he developed than is currently typical, and it
made him a natural Bayesian in statistics without him actually making the
distinction himself. 

In 1927 Jordan established the following
general formula on the number of occurrences in a sequence of events
$A_1$, $A_2$, $\ldots$, $A_n$. Set $S_0=1$ and, for $k \geq 1$,
\[ S_k = \sum_{1\leq i_1<\ldots <i_k \leq n} P\left(A_{i_1} \cap A_{i_2} \cap
\ldots \cap A_{i_k}\right). \]
Now, if $B_t$ denotes the event that exactly $t$ of the
$A_j$ occur, then
\[ P(B_t) = \sum^n_{k=t}\,\, (-1)^{k-t} \left( \begin{array}{c} k \\ t \end{array} 
\right) S_k, \qquad t = 0,1,\ldots,n. \]
The above formula now bears Jordan's name. The proof requires only
that probability is an additive set function.
In the original proof 
it is implicit that one gets upper or
lower bound above if one stops in the summation on the right hand side
at an even or odd integer, respectively. Jordan used his formula for
obtaining frequency distributions both for independent and dependent
repetitions and for combinatorial formulas associated with urn
models. He could not foresee the large literature around his
formula; see the book by J.~Galambos and I.~Simonelli (1996).

Another formula that bears his name is an interpolation
polynomial. Given that some statistical observations such as
meteorological data are measured at equidistant time (daily, weekly,
or monthly), he generated a polynomial $f(x)$ whose values are given at
points $a$, $a+h$, $a+2h$, $\ldots$, $a+nh$. The formula is easy for
computation and was very well received by contemporary scientists. He
also used the method of least squares to fit polynomials to data. If
the approximating polynomial is given by a series expansion of certain
orthogonal polynomials then it turns out that the coefficients of the
orthogonal terms in the least square polynomial do not depend on the
degree of the polynomial, hence tables are easily constructed for
computation.
Both the interpolation formula and the least square approximation
polynomials are expressible as binomial coefficients which in turn can
be generated by using finite differences. This led Jordan to study
the theory of finite differences in general, not just as a part of
statistical approximations. His book on this subject is still widely
used.

Jordan published over 90 scientific papers and books. 
A full list can be
found in the two Obituaries by 
Tak\'acs and by Gyires listed below.
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\noindent {\bf References}

\noindent J.~Galambos and I.~Simonelli (1996). {\it Bonferroni-type 
Inequalities
with Applications}. Springer Verlag, New York, NY.

\noindent B.~Gyires (1975). Jordan K\'aroly \'elete \'es munk\'ass\'aga (The
life and work of K\'aroly Jordan). {\em Alkalmazott Matematikai Lapok\/} {\bf
1}, 275-298.

\noindent L.~Tak\'acs (1961). Charles Jordan, 1871-1959. {\em Annals of
Mathematical Statistics\/}
{\bf 32}, 1-11.
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\hfill{Janos Galambos}


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