\documentclass{article} \usepackage{amsmath} \usepackage{times} % Long s % \newcommand{\s}{\mbox{$\int$}} \newcommand{\s}{s} % Ligatures with long s \newcommand{\sa}{\s a} \newcommand{\se}{\s e} \newcommand{\sh}{\s h} \newcommand{\si}{\s i} \newcommand{\so}{\s o} \newcommand{\sq}{\s q} \newcommand{\sls}{\s\s} \newcommand{\sos}{\s s} \newcommand{\st}{\s t} \newcommand{\su}{\s u} % Ligature ct \newcommand{\ct}{ct} \begin{document} \thispagestyle{empty} \noindent\hangindent=1em {\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.] \pagebreak \begin{center} {\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} \end{flushright} \end{document} %