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A 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 . . .

Combinations

Permutations

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 probability.

Today de Méré's question is taught in elementary school, but in his time had not been considered. 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 the equals of astronomers and religious leaders. Mathematics was a subject that could 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

has not changed and the coefficients for binomial expansion remain useful.

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Jacob Bernoulli's passion, however, was in curves and calculus.  One curve bears his name, the lemniscate of Bernoulli.  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 he asked with the Latin inscription, "Eadem mutata resurgo", or "Though changed, I arise again the same."

In addition to the mathematicians on the 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."

 Cardano (1501 - 1576) Pierre de Fermat (1601 - 1665) Blaise Pascal (1623 - 1662) Christiaan Huygens (1629 - 1695) Jacob Bernoulli (1654 - 1705) Editor of  Acta Eruditorum Author of Ars Conjectandi DeMoivre (1667 - 1754) Laplace (1749 - 1827)
The Ars Conjectandi (1713) images are reproduced with permission of The Huntington Library, San Marino, CA.  Students of mathematics are most grateful for the opportunity to view the original  sources.

Shirley B. Gray, February 1, 2010