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  \textbf{Abraham de Moivre}
\end{center}

    IN interesting and valuable communications to \textit{Biometrika},
of the final issues for 1924 and 1925, Karl Pearson set forth certain
facts (not all new) which will doubtless result in Abraham de Moivre
occupying a more important place than before in the history of
mathematics.  The results are reached by a careful study of (\textit{a})
``James Bernoulli's Theorem, and (\textit{b}) a publication of Moivre
dated November 12, 1733.  Regarding this publication Pearson makes
certain statements which require comment, since from them wrong
inferences might readily be drawn.

    This publication is entitled: ``Approximatio ad Summan Terminorum
Binomi $(a + b)^n$ in Seriem expansi'' and was found bound with one copy of
Moivre's ``Miscellanea Analytica,'' 1730.  Pearson remarks:

    ``Many copies of this work have attached to them a
\textit{Supplementum} with separate pagination, ending in a table of 14
figure logarithms of factorials from 10! to 900! by differences of 10. 
But only a very few copies [P.\ tells of only the one] have a second
supplement, also with separate pagination (pp.\ 1--7) and dated Nov. 12,
1733.  This second supplement could only be added to copies sold three
years after the issue of the original book, and this accounts for its
rarity.  Dr.~Todhunter in writing his \textit{History of the
Theory of Probability} appears to have used the 1730 issue of
Miscellanea Analytica, and so never came across this supplement.''

    Pearson here appears to make two slips: (\textit{a}) in assuming
that because this ``second supplement'' is bound at the end of a copy in
the University College Library, it really was a second supplement;
(\textit{b}) in stating that this publication was not considered in
Todhunter's ``History.''  Curiously enough, in the latter part of his
first article, after having indicated the results of the `second
supplement,' Pearson remarks: ``The same matter is dealt with
twenty-three years later in the edition of \textit{The Doctrine of
Chances}, pp.\ 243--250.  Todhunter in his \textit{History of the Theory
of Probability}, Arts.\ 324 and 335, passed over the topic most
superficially.''  Did Pearson overlook that a translation of his
`supplement' was thus dealt with in 1756 by Moivre and in 1865 by
Todhunter?  That this translation appeared also eighteen years before
was manifestly not recognised.

    Moreover, Moivre prefaces his translation with the statement (which
Todhunter quotes): ``I shall here translate a Paper of mine which was
printed November 12, 1733, and was communicated to some Friends, but
never yet made public, reserving to myself the right of enlarging my own
thoughts as occasion shall require.''  Hence it is clear that the
publication in question was not a 'second supplement' to Moivre's
``Miscellanea Analytica,'' but was first 'made public,' and in English,
in 1738.  A facsimile of this pamphlet is to appear in an early number
of \textit{Isis}.  It would be interesting to learn if any other copy of
Moivre's original pamphlet is in existence.
                                                                     
\begin{flushleft}
  \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
  \textsc{Raymond Clare Archibald.} \\
  \qquad Brown University, \\
  \qquad\quad Providence, R.I., \\
  \qquad\qquad February 22.
\end{flushleft}

\newpage

    I am glad that Prof. Archibald's letter enables me to return to the
subject of De Moivre's claim to be the first discoverer of the normal
curve of errors usually attributed to Laplace or Gauss.

    I do not think from what Prof. Archibald has written that he can
have seen what he thinks -- and possibly may be -- the unique copy of the
``Approximatio ad Summam Terminorum Binomi.''  It is in the same type
and has the same characteristic species of tailpiece as the
``Miscellanea Analytica.''  It has the same unusual form of pagination
as the first ``Supplementum''; it is printed on the same paper and is of
the same format as the first ``Supplementum'' and the ``Miscellanea.'' 
About half the known copies of the ``Miscellanea Analytica'' have not
the first supplement, and I think it quite probable that the
``Approximatio'' was only bound up with a few last copies of the
``Miscellanea Analytica'' issued after November 1733.  At any rate, that
is where I should seek for it first.  I said in my paper that De Moivre
treated of the same subject in his ``Doctrine of Chances,'' 1756,
because that was the edition I was working with.  Prof. Archibald says
that it was also treated of in the 1738 edition, and then speaks as if
the matter in the 1738 ``Doctrine of Chances'' and again in the 1756
edition was a mere translation ``except for minor changes.''  This is
not correct; for the history of statistics most important additions were
made in both the 1738 and the 1756 editions.  The important principle of
the `activating deity' maintaining the stability of statistical ratios
does not appear in the 1733 ``Approximatio''; it first appears
tentatively in the 1738 ``Doctrine,'' where the seven lines of Corollary
X are increased to nearly fifty lines, while in the 1756 ``Doctrine''
this corollary alone occupies some four pages or about 160 lines. 
Indeed the 6 pages of the ``Approximatio'' becomes $11\frac{1}{2}$
pages (of more lines) in the 1756 ``Doctrine.''  As De Moivre
appropriately observes, he has reserved to himself ``the right of
enlarging my own thoughts.''  That enlargement, developing Newton's idea
of an omnipresent activating deity, who maintains mean statistical
values, formed the foundation of statistical development through Derham,
S\"ussmilch, Niewentyt, Price to Quetelet and Florence Nightingale.  These
may be mathematically `minor' points, but they are vital for the history
of statistics, and my reference to these additions in the penultimate
paragraph of my paper might have shown Prof. Archibald that I was aware
of the differences between the original ``Approximatio'' and the same
dealt with in the ``Doctrine of Chances.''  My error lay in not
recognising that in the 1738 ``Doctrine,'' the $6\frac{1}{2}$ pages had
grown to a little over 8, to become $11\frac{1}{2}$ pages eighteen years
later.

    As to Dr. Todhunter, I have nothing whatever to retract in my
judgement.  In his Art. 335 he misses entirely the epoch-making
character of the ``Approximatio'' as well as its enlargement in the
``Doctrine.''  He does not say ``Here is the original of Stirling's Theorem,
here is the first appearance of the normal curve, here De Moivre
anticipated Laplace as the latter anticipated Gauss.  He does not even
refer to the manner in which De Moivre expanded the Newtonian theology
and directed statistics into the channel down which it flowed for nearly
a century.  Almost everywhere in his ``History'' Todhunter seizes a small
bit of algebra out of a really important memoir and often speaks of it
as a school exercise, whereas the memoir may have exerted by the
principles involved a really wide influence on the development of the
mathematical theory of statistics, and ultimately on statistical
practice also.

    Todhunter fails almost entirely to catch the drift of scientific
evolution, or to treat that evolution in relation to the current thought
of the day, which influences science as much as science influences
general thought.  The causes which led De Moivre to his ``Approximatio''
or Bayes to his theorem were more theological and sociological than
purely mathematical, and until one recognizes that the post-Newtonian
English mathematicians were more influenced by Newton's theology than by
his mathematics, the history of science in the eighteenth century -- in
particular that of the scientists who were members of the Royal Society
-- must remain obscure.\qquad\qquad\textsc{Karl Pearson.}

\bigskip

    March 18, 1926.

\bigskip\bigskip

\noindent
\textit{Nature} \textbf{117} (1926 April 17), 551--552.

\newpage

\begin{center}
  {\Large\textbf{A Rare Pamphlet of Moivre and some of his Discoveries}}
\end{center}

    As a result of two recent studies by \textsc{Karl
      Pearson}\footnote{K. Pearson, `` Historical note on the origin of
      the normal curve of errors, '' \textit{Biometrika}, vol. 16, pp.\
      402--404, Dec., 1924; `` \textsc{James Bernoulli}'s Theorem, ''
      \textit{Biometrika}, vol. 17, pp.\ 201--210, Dec., 1925.} valuable
information is at hand for forming a true estimate of the important
position which \textsc{Abraham de Moivre} occupies in the history of
mathematics.  These studies are based on a consideration of : (a) part 4
of \textsc{Jean Bernoulli}'s \textit{Ars Conjectandi}\footnote{Published
  at Basle in 1713, eight years after \textsc{Bernoulli}'s death, by his 
  nephew \textsc{Nicolas Bernoulli}.  Pearson shows that \flqq\ Pars Quarta 
  \frqq\ of the \textit{Ars Conjectandi} has not the importance which has often 
  been attributed to it.}, which contains what is properly called  \flqq\
Bernoulli's Theorem \frqq; and (b) a pamphlet of \textsc{Moivre} dated
November 12, 1733.

    The chief purpose of this paper is to summarize some of
\textsc{Pearson}'s results, to remove misapprehension which one of his
pamphlets may cause, and to present a facsimile of the \textsc{Moivre}
pamphlet which is entitled : \textit{Approximatio ad Summam Terminorum
Binomii $(a+b)^n$ in Seriem expansi.}

    \textsc{Pearson} opens his article as follows :

    \flqq\ It is usual to attribute the discovery of the Normal curve of
errors to \textsc{Gauss}.  This is solely due to the fact that 
\textsc{Laplace}'s \textit{Th\'eorie analytique des Probabilit\'es} was
published in 1812, and to this most writers have referred.  But
\textsc{Laplace}'s \textit{M\'emoire sur les Probabilit\'es} was
published in the \textit{Histoire de l Acad\'emie des Sciences} in 1778,
and this memoir contains the normal curve function, and emphasizes the
importance of tabulating the probability integral.  Nay, we go further
and say that (\textit{M\'emoires . . . pr\'esent\'es} \textsc{T.~iv.}\ 
p.~6) when discussing \textsc{Bayes}' Theorem had also reached the
exponential curve of errors as an approximation to the hypergeometrical
series.  All \textsc{Gauss}' work falls in the 19th century.  His
\textit{Theoria motus corporum coelestium} was published in 1809, his
theory of least squares and his theory of combination of observations
being of a still later date.  There is, I think, not a doubt that
\textsc{Laplace}'s name ought to be associated with the normal curve and the
probability integral before \textsc{Gauss}'.

      \glqq\ But in studying \textsc{De Moivre} I have come across a work 
which long antedates both \textsc{Laplace} and \textsc{Gauss}.

      \glqq\ The matter is a very singular one historically.  \textsc{De
Moivre} published in 1730 his \textit{Miscellanea Analytica}, still a
mine of hardly fully explored wealth.  Many copies of this work have
attached to them a \textit{Supplementum} with separate pagination,
ending in a table of 14 figure logarithms of factorials from 10! to 900!
by differences of 10.  But only a \textit{very few} copies
[\textsc{Pearson} found only one!] have a second supplement, also with
separate pagination (pp.~1--7) and dated Nov.\ 12, 1733.  This second
supplement could only be added to copies sold three years after the
issue of the original book, and this accounts for its rarity. 
Dr.~\textsc{Todhunter} in writing his \textit{History of the Theory of
Probability} appears to have used the 1730 issue of \textit{Miscellanea
Analytica}, and so never come across this supplement. \grqq  

    In this last paragraph \textsc{Pearson} makes statements which might
readily mislead : firstly, in suggesting that because this \flqq\ second
supplement \frqq\ is bound at the end of a copy of \textit{Miscellanea
Analytica} in the University College, London, it really was a supplement
; secondly, in asserting that this publication was not considered in
\textsc{Todhunter}'s \textit{History}.  As a matter of fact this \flqq\
second supplement \frqq\ appeared in English, except for minor changes (see
below), in both the second and third editions of \textsc{Moivre}'s
\textit{Doctrine of Chances}\footnote{Second edition, London, 1738, pp.\
  235--242; third edition, 1756, pp. 243--250 after translation (pp.
  250--254, 334) is interesting new material.}.  \textsc{Todhunter}
considers this in paragraphs 324 and 335 of his
History\footnote{Curiously enough in the latter part of his first
  article, after having indicated the results of the \flqq\ second supplement
  \frqq, \textsc{Pearson} remarks : \flqq The same matter is dealt with 
  twenty-three years later in the edition of 1756 of the \textit{Doctrine of 
  Chances}, pp.\ 243--250.  \textsc{Todhunter} in his \textit{History of the 
  Theory of Probability}, Arts, 324 and 335, passed over this topic most
  superficially.\frqq\  Did Pearson overlook it was a translation of his \flqq\ 
  supplement \frqq\  that was dealt with in 1756 by \textsc{Moivre}, and in
  1865 by \textsc{Todhunter}?  That this translation appeared also 18 years 
  before was manifestly not recognized in the articles in question.}.  Moreover
\textsc{Moivre} prefaces his translation with the statement :  I shall here
translate a Paper of mine which was printed \textit{November} 12, 1733, and
communicated to some Friends, but never yet made public, reserving to
myself the right of enlarging my own Thoughts ,[\footnote{[In
    the 1738 translation there is practically no change from the
    original before corollary 4, where there is a slight alteration in
    the first clause ; corollary 5 is more than doubled in length by the
    addition of a third paragraph ; additions have been made near the
    first and last of corollary 6 and the odds `` 792 ad 1 proxime '' have
    been changed to ``369 to 1 nearly ''; by additions lemma 2 has been
    more than trebled in length; corollary 9 is slightly, but corollary
    10 greatly, extended.]}] as occasion shall require.  Hence it is
clear that the publication in question was not a \flqq\ second supplement 
\frqq\ to \textsc{Moivre}'s \textit{Miscellanea Analytica} but was first 
\flqq\ made public \frqq\ in 1738.
 Another copy of Moivre's original pamphlet is in the Preussiche
Staatsbibliothek, Berlin.

    \textsc{Pearson} gives priority to \textsc{Moivre} in :
\begin{enumerate}
  \item Presenting the first treatment of the probability integral, and
essentially of the normal curve;
  \item Formulating and using the theorem, improperly called Stirling's
theorem;
  \item Enunciating the theorem that the measure of accuracy depends on
the inverse square root of the size of the sample, so often called
\textsc{Bernoulli}'s theorem\footnote{A correct statement of Bernoulli's
  theorem, given on page 236, of \textit{Ars Conjectandi}, is as follows : Sit
  igitur numerus casuum fertilium ad numerum sterilium vel pr\ae cis\`e,
  vel proxim\`e in ratione $r/s$, adeoque ad numerum omnium in ratione
  $r/(r+s)$ seu $r/t$, quam rationem terminent limites $(r+1)/t\ \ (r-1)/t$.  
  Ostendum est, tot posse capi experimenta, ut datis quotliber (puta $c$) 
  vicibus, versimilius evadut, numerum fertilium observationum intra hos 
  limites qu\`am extra casurum esse, h.e.\ numerum fertilium ad numerum omnium
  observationum rationem habiturum nec majorem qu\`am $(r+1)/t$, nec minorem
  qu\`a $(r-1)/t$.  [Therefore, let the number of fertile cases to the
  number of sterile cases be exactly or approximately in the ratio $r$ to
  $s$, and hence the ratio of fertile cases to all the cases will be
  $r/(r+s)$ or $r/t$, which is within the limits $(r+1)/t$ and $(r-1)/t$.  It
  must be shown that so many trials can be run such that it will be more
  probable than any given times (e.g., $c$ times) that
  the number of fertile observations will fall within these limits
  rather than outside these limits -- i.e., it will be $c$ times more
  likely than not that the number of fertile observations to the number
  of all the observations will be in a ratio neither greater than
  $(r+1)/t$ nor less than $(r-1)/t$.]} although it is entirely due to 
  \textsc{Moivre}.
\end{enumerate}

    Moreover for this theorem \textsc{Moivre} appreciated the immense
range of application.  \flqq\ For him it was a theological problem, he was
determining the frequency of irregularities from the Original Design of
the Deity.  Without grasping this side of the matter, it is impossible
to understand the history of statistics from \textsc{De Moivre} through
\textsc{Derham} and \textsc{S\"ussmilch} to \textsc{Quetelet},
culminating in the modern principle of the stability of statistical
ratios.  No one had any true inkling of the ideas of probable deviation
of the statistical ratio before \textsc{De Moivre} \frqq.

    Pearson concludes his second paper as follows : \flqq\
\textsc{Bernoulli} saw the importance of a certain problem ; so did
\textsc{Ptolemy}, but it would be rather absurd to call
\textsc{Kepler}'s or \textsc{Newton}'s solution of planetary motion by
\textsc{Ptolemy}'s name!  Yet an error of like magnitude seems to me
made when \textsc{De Moivre}'s method is discussed without reference to
its author, under the heading of \flqq\ \textsc{Bernoulli}'s Theorem \frqq. 
The contribution of the \textsc{Bernoullis} to mathematics is
considerable, but they have been in more than one instance greatly
exaggerated.  The \textit{Pars Quarta} of the \textit{Ars Conjectandi}
has not the importance which has been attributed to it. \frqq

    Now as to a second theorem for which \textsc{Moivre} deserves
credit, namely the one wrongly associated with the name of
\textsc{Stirling}, \textsc{Moivre} finds the ratio of the maximum terms
of a binomial to the term at a distance x from the maximum.

    \flqq He supposes his power so large, that we may practically use
\[ m! = \textit{const.}\times\sqrt{m}\, e^{-m} m^m. \]
He determines the constant which he calls $B$ by the theorem that the
hyperbolic logarithm of $B$ is given by :
\[ \log_e B = 1 - 1/12 + 1/360 - 1/1260 + 1/1680,\text{\ etc.}, \]
which gives $\log B = .399,2235$ to the terms written down, or $B = 2.5074$.

    Thus
\[ m! = 2.5074 \times \sqrt{m}\, e^{-m} m^m.\text{\ \frqq} \]

    Then \textsc{Moivre} states : \flqq\ When I first began that inquiry, I
contented myself to determine at large the Value of $B$, which was done
by the addition of some Terms of the above-written Series; but as I
perceiv'd that it converged but slowly, and seeing at the same time that
what I had done answered my purpose tolerably well, I desisted from
proceeding further, till my worthy and learned Friend Mr. \textsc{James
Stirling}, who had applied himself to that inquiry, found that the
quantity $B$ did denote the Square-root of the Circumference of a Circle
whose radius is unity \frqq\ \textsc{Pearson} remarks very rightly : \flqq\ I
consider that the fact that \textsc{Stirling} showed that \textsc{De
Moivre}'s arithmetical constant was $\sqrt{2\pi}$ does not entitle him
to claim the theorem, and it is erroneous to term it \textsc{Stirling}'s
Theorem [\footnote{\textsc{J.~Stirling}, \textit{Methodus
  Differentialie}, London 1730, p.\ 137 ; English edition by F.
  Holliday, London, 1749, p. 121.  While \textsc{Stirling} gave, in 
  effect, the formula, $a$ being the reciprocal of \textsc{Napier}'s 
  logarithm of 10 :
  \[ \text{Log\,}x! = \frac{1}{2}\log 2\pi + (x+\frac{1}{2})\log(x+\frac{1}{2})
               -(x+\frac{1}{2})a-\frac{a}{2.12.(x+\frac{1}{2})}
               +\frac{a}{8.360\,(x+\frac{1}{2})^3} \]
  with the law for the continuation of the series, Moivre expressed the
  result, in effect, in the more convenient form (compare, \textsc{C.~Tweedie},
  \textit{James Stirling}, Oxford, 1922, p.\ 119, 203--205) :
  \[ \log x! = \frac{1}{2}\log 2\pi + (x+\frac{1}{2})\log x - x
     +\frac{B_1}{1.2}\frac{1}{x}+\frac{B_3}{3.4}\frac{1}{x^3} + \dots \]
  $B_1$ , $B_3$ denoting the \textsc{Bernoulli} numbers.  But in spite
  of the fact that it was \textsc{Moivre}, and not \textsc{Stirling}, who 
  gave the series
  \[ \frac{B_1}{1.2}\frac{1}{x}+\frac{B_3}{3.4}\frac{1}{x^3} + \dots \]
  this series has been called \textsc{Stirling}'s series by Godefroy, 
  \textit{Th\'eorie \'el\'ementaire des S\'eries}, Paris, 1903, pp.\ 
  224--228, and others.  Compare \textit{Encylop\"adie der mathematischen
  Wissenschaften}, vol.\ 1, part 2, 1900--1904, p.\ 931.}]

    A facsimile copy of the original pamphlet follows\footnote{I am
      indebted to the Librarian of University College, London, for his
      courtesy in allowing a photostatic copy to be made.  Should any
      reader know of other copies of the original, he would confer a
      favor on the writer by informing him where they may be found.}.

\textit{(Brown University, Providence, R.\ I.)}
\hfill\textsc{R.\ C.\ Archibald}

\bigskip

\noindent
\textit{Isis} \textbf{8} (1926), 671--683.

\bigskip\bigskip

\noindent \textit{[The text of the {\em Supplementum} can be found in 
  A~De~Moivre, {\em The Doctrine of Chances} (2nd ed.), London: H~Woodfall
  1738, reprinted London: Cass 1967, or A~De~Moivre, {\em The Doctrine of 
  Chances} (3rd ed.), London: A~Millar 1756, reprinted New York, NY: 
  Chelsea 1967 with a biographical article from {\em Scripta Mathematica} 
  \textbf{2} (4) (1934), 316-333, by H~M~Walker.  It is also reprinted
  in  D~E~Smith, {\em A Source Book in Mathematics}, 2 vols, New York, 
  NY: McGraw-Hill 1929, reprinted New York, NY: Dover 1959, and in 
  Appendix 5 of F~N~David, {\em Games, Gods and Gambling}, London: 
  Griffin 1962.]}

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