216 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICSnumber of objects chosen. The total number of permutations of a number of objects
he called its variationes, and for the number of combinations of a set of objects
taken, say, four at a time, he wrote confaatio, an abbreviation for conquattuornatio.
The case of two objects taken at a time provides the modern word combination.
These combinations, now called binomial coefficients, were referred to generically as
complexiones. The first problem posed by Leibniz was: Given the numerus and the
exponent, find the complexiones. In other words, given ç and fc, find the number
of combinations of ç things taken k at a time.
Like the Hindu mathematicians, Leibniz applied combinatorics to poetry and
music. He considered the hexameter lines possible with the Guido scale ut, re, mi,
fa, sol, la, finding a total of 187,920.^18
De arte combinatoria contains 12 sophisticated counting problems and a num-
ber of exotic applications of the counting techniques. It appears that Leibniz in-
tended these techniques to be a source by which all possible propositions about
the world could be generated. Then, combined with a good logic checker, this
technique would provide the key to all knowledge. His intent was philosophical as
well as mathematical, as evidenced by his claimed mathematical proof of the exis-
tence of God at the beginning of the work. Thus once again, this particular area
of mathematics seems to be linked, more than other kinds of mathematics, with
mysticism. The frontispiece of De arte combinatoria shows a mystical arrangement
of the opposite pairs wet/dry, cold/hot, with the four elements of earth, air, fire,
and water as cardinal points. This figure resembles an elaborate version of the fa-
mous ying/yang symbol from Chinese philosophy and it also recalls the proposition
generator of the mystic theologian Ramon Lull (1232-1316), which consisted of a
series of nested circles with words inscribed on them. When rotated independently,
they would generate sentences. Leibniz was familiar with Lull's work, but was not
a proponent of it.
The seeds planted by Leibniz in De arte combinatoria sprouted and grew dur-
ing the nineteenth century, as problems from algebra, probability, and topology
required sophisticated techniques of counting. One of the pioneers was the British
clergyman Thomas Kirkman (1806-1895). The first combinatorial problem he
worked on was posed in the Lady's and Gentleman's Diary in 1844: Determine
the maximum number of distinct sets of ñ symboh that can be formed from a set
of ç symbols subject to the restriction that no combination of q symboh can be
repeated in different sets. Kirkman himself posed a related problem in the same
journal five years later: Fifteen young ladies in a school walk out three abreast for
7 days in succession; it is required to arrange them daily so that no two shall walk
twice abreast. This problem is an early example of a problem in combinatorial de-
sign. The problem of constructing a Latin square is another example. This kind of
combinatorial design has a practical application in the scheduling of athletic tour-
naments, and in fact colleagues of the author specializing in combinatorial design
procured a contract to design the schedule for the short-lived XFL Football League
in 2001.
We shall terminate our discussion of combinatorics with these nineteenth-
century results. We note in parting that it remains an area with a plenitude of
unsolved problems whose statement can be understood without long preparation.
(^18) The first five of these tones are the first syllables of a medieval Latin chant on ascending tones.
The replacement of ut by the modern do came later.