Lemniscate of Bernoulli

Permutations

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.

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.

_______________________

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."

Pierre de Fermat
(1601 - 1665)



Blaise Pascal
(1623 - 1662)



Christiaan Huygens

(1654 - 1705)
Editor of  Acta Eruditorum
Author of Ars Conjectandi



DeMoivre

Laplace