Primer on Early Probability
James (Jacob, Jacques)
the oldest of the famed Bernoulli mathematicians
From the 1713 volume of "Ars
Conjectandi" published eight years after his death . . .
In reading the Italian publications of Pacioli's Summa,
(1494), Cardano's Liber de Ludo Aleae (1526) and
Trataglia's Generale Trattato ( 1556-1560 ), one finds
examples of how to win at various games of chance.
Indeed, throughout civilization anthropologists have found evidence of
games that most likely involved gambling.
Traditionally, however, historians of
mathematics have written that probability as a discipline began with a
triumvirate of Frenchmen, the Chevalier de Méré, Blaise
Pascal and Pierre de Fermat. De Méré
asked Pascal how
many throws should be allowed to provide even odds for rolling two
sixes on a pair of dice. Intrigued, Pascal passed along this question
and others in letters to Fermat. Other than an exchange of letters,
none published any conclusions.
Though brief, Christiaan Huygens' De
Ratiociniis in Aleae Ludo
or On the Calculations
in Games of
Chance (1657) is now
considered the first text written on
is taught in elementary school, but in his time had not been
Pascal's Triangle is one of the most famous arrays in all mathematics.
Fermat is remembered not for probability, but for his lasting comment
on finding a proof for a theorem whose solution perplexed the most
talented mathematicians for 450 years.
It remained for the special talents of
James Bernoulli to unite the fragmented findings of these men into an
emerging field of truly great mathematics. By editing, refining and
polishing their works in Acta Eruditorum
along with those of Newton, Leibniz and a host of other
philosophers, mathematicians presented themselves to the world as truly
equals of astronomers and religious leaders. Mathematics was a subject
not be ignored among the educated. Bernoulli's notes on
probability were published posthumously in Ars
Conjectandi (1713). Near the end is found his Law of Large Numbers and Bernoulli numbers.
This particular web page invites you
to see Bernoulli's calculation for permutations. (Factorial notation
had not appeared.) Note, however, the basic arithmetic of "6 choose
6" arrangements or
changed and the coefficients for binomial expansion remain useful.
Jacob Bernoulli's passion, however, was in curves and calculus.
One curve bears his name, the lemniscate of
. In particular, the logarithmic spiral captured
his attention to the point
that he requested it be engraved on his tomb! On
visiting the cathedral in Basel, Switzerland, high above the Rhine, one
cannot help but be impressed by his prominent sepulcher adorned just as
asked with the Latin inscription, "Eadem mutata resurgo", or "Though
changed, I arise again the same."
In addition to the mathematicians on
right, we must remember Thomas Bayes (1702-1761) from across the
English Channel as making outstanding contributions to
probability. His famous
theorem was a direct result of interaction with the findings of others,
especially Bernoulli and De Moivre. Classically he wrote, "Given
the number of times in which an unknown event has happened and failed,
. . . the probabiilty of its happening in a single trial lies somewhere
between any two degrees of probability that can be named."