- PROBABILITY 517
1.6. De Moivre. In 1711, even before the appearance of Jakob Bernoulli's trea-
tise, another groundbreaking book on probability appeared, the Doctrine of Chances,
written by Abraham de Moivre (1667-1754), a French Huguenot who took refuge
in England after 1685, when Louis XIV revoked the Edict of Nantes, which had
guaranteed civil rights for Huguenots when Henri IV took the French throne in1598.^7 De Moivre's book went through several editions. Its second edition, which
appeared in 1738, introduced a significant piece of numerical analysis, useful for ap-
proximating sums of terms of a binomial expansion (a + b)n for large n. De Moivre
had published the work earlier in a paper written in 1733. Having no notation
for the base e, which was introduced by Euler a few years later, de Moivre simply
referred to the hyperbolic (natural) logarithm and "the number whose logarithm
is 1." De Moivre first considered only the middle term of the expansion. That is,
for an even power ç = 2m, he estimated the term
'2mA (2m)!
m J (m!)^2
and found it equal to g^j, where  was a constant for which he knew only an
infinite series. At that point, he got stuck, as he admitted, until his friend James
Stirling (1692-1770) showed him that "the Quantity  did denote the Square-root
of the Circumference of a Circle whose Radius is Unity." In our terms, Â = V2n,
but de Moivre simply wrote c for B. Without having to know the exact value of Â
de Moivre was able to show that "the Logarithm of the Ratio, which a Term distant
from the middle by the Interval I, has the the middle Term, is [approximately, for
large n] In modern language,
De Moivre went on to say, "The Number, which answers to the Hyperbolic Loga-
rithm -2ll/n, [is]
l_2U AP__^_ m&
32*^10 64Z^12
ç + 2nn 6n3 + 24ç^4 120n5 + 720n^7 ' °'
By scaling, de Moivre was able to estimate segments of the binomial distribution.
In particular, the fact that the numerator was I^2 and the denominator ç allowed
him to estimate the probability that the number of successes in Bernoulli trials
would be between fixed limits. He came close to noticing that the natural unit of
probability for ç trials was a multiple of yfri. In 1893 this natural unit of measure
for probability was named the standard deviation by the British mathematician
Karl Pearson (1857-1936). For Bernoulli trials with probability of success ñ at
each trial the standard deviation is ó = y/np(l - p).
For what we would call a coin-tossing experiment in which ñ — \ —he imagined
tossing a metal disk painted white on one side and black on the other—de Moivre
observed that with 3600 coin tosses, the odds would be more than 2 to 1 against
a deviation of more than 30 "heads" from the expected number of 1800. The
standard deviation for this experiment is exactly 30, and 68 percent of the area
under a normal curve lies within one standard deviation of the mean. De Moivre(^7) The spirit of sectarianism has infected historians to the extent that Catholic and Protestant
biographers of de Moivre do not agree on how long he was imprisoned in France for being a
Protestant. They do agree that he was imprisoned, however. To be fair to the French, they did
elect him a member of the Academy of Sciences a few months before his death.