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{\Large XLIII. \textit{A letter from the late Reverend Mr.}\ Thomas Bayes,
\textit{F.\ R.\ S.\ to} John Canton, \textit{M.\ A.\ and F.\ R.\ S.}}
\bigskip
{\large SIR,}
\bigskip
\noindent
{\footnotesize Read Nov.\ 24, 1763.}
If the following ob{\se}rvations do not {\se}em to you to be too minute, I
{\sh}ould e{\st}eem it as a favour, if you would plea{\se} to communicate
them to the Royal Society.
It has been a{\sls}erted by {\so}me eminent mathematicians, that the {\su}m of
the logarithms of the numbers 1\,.\,2\,.\,3\,.\,4 \&c to $z$, is equal to
$\frac{1}{2}\log.\, c +\overline{z+\frac{1}{2}}\times\log.\, z$
le{\sls}ened by the {se}ries
$z-\frac{1}{12z}+\frac{1}{360z^3}-\frac{1}{1260z^5}+\frac{1}{1680z^7}
-\frac{1}{1182z^9}+\text{\&c.}$
if $c$ denote the circumference of a circle who{\se} radius is unity. And
it is true that this expre{\sls}ion will very nearly approach to the value
of that {\su}m when $z$ is large, and you take in only a proper number of
the fir{\st} terms of the foregoing {\se}ries; but the whole {\se}ries can
never properly expre{\sos} any quantity at all; becau{\se} after the 5th term
the coefficients begin to increa{\se}, and they afterwards increa{\se} at a
greater rate than what can be comprehended by the increa{\se} of the powers
of $z$, though $z$ repre{\se}nt a number ever {\so} large; as will be evident
by con{\si}dering the following manner in which the coefficients of that
{\se}ries may be formed. Take $a=\frac{1}{12}$, $5b=a^2$, $7c=2ba$,
$9b=2ca+b^2$, $11e=2da+2cb$, $13f=2ea+2db+c^2$, $15g=2fa+2eb+2dc$, and
{\so} on; then take $\text{A}=a$, $\text{B}=2b$, $\text{C}=2\times3\times4c$,
$\text{D}=2\times3\times4\times5\times6d$,
$\text{E}=2\times3\times4\times5\times6\times7\times8e$ and {\so} on, and
A, B, C, D, E, F, \&c. will be the coefficients of the foregoing {\se}ries
: from whence it ea{\si}ly follows, that if any term in the {\se}ries after
the 3 fir{\st} be called $y$, and its di{\st}ance from the fir{\st} term $n$,
the term immediately following will be greater than
$\frac{n\times\overline{2n-1}}{6n+9}\times\frac{y}{z^2}$.
Whereforeatlength the {\su}b{\se}quent terms of this {\se}ries are greater than
the preceding ones, and increa{\se} in infinitum, and therefore the whole
{\se}ries can have no ultimate value what{\so}ever.
Much le{\sos} can that {\se}ries have any ultimate value, which is deduced
from it by taking $z=1$, and is {\su}ppo{\se}d to be equal to the logarithm of
the {\sq}uare root of the periphery of a circle who{\se} whole radius is unity;
and what is {\sa}id concerning the foregoing {\se}ries is true, and appears to
be {\so}, much in the {\sa}me manner, concerning the {\se}ries for finding out
the {\su}m of the logarithms of the odd numbers 3\,.\,5\,.\,7 \&c.\dots$z$ and
tho{\se} that are given for finding out the {\su}m of the infinite
proggre{\sls}ions, in which the {\se}veral terms have the {\sa}me numerator
whil{\st} their denominators are any certain power of numbers increa{\si}ng in
arithmetical proportion. But it is needle{\sos} particularly to in{\si}{\st}
upon the{\se}, becau{\se} one in{\st}ance is {\su}fficient to {\sh}ew that
tho{\se} methods are not to be depended upon, form which a conclu{\si}on
follows that is not exa{\ct}.
\bigskip\bigskip
\noindent
[\textit{Philosophical Transactions of the Royal Society of London}
\textbf{53} (1763), 269--271.]
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{\large SOME REMARKS CONCERNING BAYES' NOTE ON THE USE OF CERTAIN
DIVERGENT SERIES}
\end{center}
The same volume of the Philosophical Transactions (1763) that contains
Bayes' essay ``towards solving a problem in the doctrine of chances,''
contains also a note on divergent series, barely covering two pages, yet
describing the essentials, save for a discussion of the remainder term,
of what we now call asymptotic expansions. At the foot of page 401 of
the essay Price refers to a note by Bayes on series, and it is this
note that we now have in mind. The occasion was the divergent series
for the factorials in the term $Ea^pb^q$, or in more familiar notation,
the term $\binom{n}{q}a^{n-q}b^q$ in the binomial expansion of
$(a+b)^n$. For his illustration Bayes dealt with the series
\begin{align*}
\log z! &\sim\log\sqrt{2\pi}+(z+\mbox{$\frac{1}{2}$})\log z \\
&\qquad\quad -\left\{z-\frac{1}{1.2z}B_1+\frac{1}{3.4z^3}B_3
-\frac{1}{5.6z^5}B_2+-\dots\right\}
\end{align*}
commonly known as Stirling's formula, and used by DeMoivre in his
derivation of the binomial (1733). It seems more probable that Bayes
had DeMoivre and Stirling in mind when he spoke of ``some eminent
mathematicians,'' as several clues indicate: \textit{first,} the symbol
$c$ for $2\pi$ was commonly used by DeMoivre and Stirling;
\textit{second,} of all the series that Bayes could have chosen for his
illustration, the one he actually used in the one to which these two men
had devoted so much study; \textit{third,} Price, who knew Bayes' mind
as well as anyone, speaks of Stirling on page 401 of the essay with the
same adjective; \textit{fourth,} in the \textit{Approximatio} DeMoivre
in the words \textit{attamen tarditas convergientiae me deterruerat
quominus longius procedurem} relates his struggle with the series
\[ B=-\frac{1}{12}+\frac{1}{360}-\frac{1}{1260}+\frac{1}{1680}+\text{\&c.} \]
which is none other than the result of putting $z=1$ in the
complementary terms of the expansion for $\log z!$.
\textit{Tarditas convergientae:} apparently neither he nor Stirling
realized that his series is only symbolic for $\log\sqrt{2\pi}$, and
that it does not converge at all. If they had written down one more
term (1/1188) they would have perceived \textit{divergence}. But we
must remember that convergence was not carefully defined as a limiting
process in those days, though Stirling in his \textit{Methodus
Differentialis} gave the earliest test for convergence.
At the time Bayes dies, Euler's \textit{Institutiones Calculi
Differenitalis} had been in existence six years. Therein on page 465
Euler remarks on the divergence of the factorial series for $z=1$,
whereupon for the evaluation of the constant ($\log\sqrt{2\pi}$) he
resorts to Wallis' product, as Stirling had done before him. It is not
possible to judge from this or his earlier writings whether Euler was
aware that this series diverges for all values of $z$ however large;
neither is it clear whether he recognised the eventual divergence of the
``Euler-Maclaurin summation formula'' for functions not polynomials. It
may therefore be that this note by Bayes contains the first recognition
of the existence of an optimum term, i.e., of the asymptotic character
of the series, and one can not help but wonder when he actually wrote
it; in particular, whether he perceived it from anything that Euler had
said. The manuscript that Price sent in to the Royal Society is not
dated, and whether Price sent Bayes' original paper to the Royal
Society, or copied it omitting the date, no one seems to know. Many
enquiries received by the Secretaries of the Royal Society give evidence
of curiosity before ours; but regardless of the answer, the note
commands respect for the mathematical attainments of the author of the essay.
\begin{flushright}
\textsc{W.~Edwards~Deming}
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