% LaTeX source for Barnard on Bayes







  {\large\textbf{\textsc{By} G. A. BARNARD}}



Bayes's paper, reproduced in the following papers, must rank as one of
the most famous memoirs in the history of science and the problem it
discusses is still the subject of keen controversy.  The intellectual
stature of Bayes himself is measured by the fact that it is still of
scientific as well as historical interest to know what Bayes had to say
on the questions he raised.  And yet such are the vagaries of historical
records, that almost nothing is known about the personal history of the 
man.  \textit{The Dictionary of National Biography}, compiled at the end
of the last century, when the whole theory of probability was in
temporary eclipse in England, has an entry devoted to Bayes's father,
Joshua Bayes, F.R.S., one of the first six Nonconformist ministers to
be publicly ordained as such in England, but it has nothing on this much
more distinguished son.  Indeed, the note on Thomas Bayes which is to
appear in the forthcoming new edition of the \textit{Encyclopedia
Britannica} will apparently be the first biographical note to appear in
a work of general reference since the \textit{Imperial Dictionary of
Universal Biography} was published in Glasgow in 1865.  And in treatises
on the history of mathematics, such as that of Loria (1933) and Cantor
(1908), notice is taken of his contributions to probability theory and
to mathematical analysis, but biographical details are lacking.

The Reverend Thomas Bayes, F.R.S., author of the first expression in
precise, quantitative form of one of the modes of inductive inference,
was born in 1702, the eldest son of Ann Bayes and Joshua Bayes, F.R.S. 
He was educated privately, as was usual with Nonconformists at that
time, and from the fact that when Thomas was 12 Bernoulli wrote to
Leibniz that `poor de Moivre' was having to earn a living in London by
teaching mathematics, we are tempted to speculate that Bayes may have
learned mathematics from one of the founders of the theory of
probability.  Eventually Thomas was ordained, and began his ministry by
helping his father, who was at the time stated, minister of the
Presbyterian meeting house in Leather Lane, off Holborn.  Later the son
went to minister in Tunbridge Wells at the Presbyterian Chapel on Little
Mount Sion which had opened on 1 August 1720.  It is not known when
Bayes went to Tunbridge Wells, but he was not the first to minister on
Little Mount Sion, and he was certainly there in 1731, when he produced
a tract entitled `Divine Benevolence, or an attempt to prove that the
Principle End of the Divine Providence and Government is the happiness of
His Creatures'.  The tract was published by John Noon and copies are in
Dr Williams's library and the British Museum.  The following is a


[p.\ 22]; I don't find (I am sorry to say it) any necessary connection
between mere intelligence, though ever so great, and the love or
approbation of kind and beneficent actions.


Bayes argued that the principal end of the Deity was the happiness of
His creatures, in opposition to Balguy and Grove who had, respectively,
maintained that the first spring of the action of the Deity was
Rectitude and Wisdom.

In 1736, John Noon published a tract entitled `An Introduction to the
Doctrine of Fluxions, and a Defence of the Mathematicians against the
objections of the Author of the Analyst'.  De Morgan (1860) says:
`This very acute tract is anonymous, but it was always attributed to
Bayes by the contemporaries who write in the names of the authors as I
have seen in various copies, and it bears his name in other places.' 
The ascription to Bayes is accepted also in the British Museum

From the copy in Dr Williams's library we quote:


[p.\ 9]: It is not the business of the Mathematician to dispute whether
quantities do in fact ever vary in the manner that is supposed, but only
whether the manner of their doing so be intelligible; which being
allowed, he has a right to take it for granted, and then see what
deductions he can make from that supposition.  It is not the business of
a Mathematician to show that a strait line or circle can be drawn, but
he tells you what he means by these; and if you understand him, you may
proceed further with him; and it would not be to the purpose to object
that there is no such thing in nature as a true strait line or perfect
circle, for this is none of his concern: he is not inquiring how things
are in matter of fact, but supposing things to be in a certain way, what
are the consequences to be deduced from them; and all that is to be
demanded of this is, that his suppositions be intelligible, and his
inferences just from the suppositions he makes.

[p.\ 48]: He [i.e.\ the Analyst = Bishop Berkeley] represents the
disputes and controversies among mathematicians as disparaging the
evidence of their methods: and, Query 51, he represents Logics and 
Metaphysics as proper to open their eyes, and to extricate them from their
difficulties.  Now were ever two things thus put together?  If the
dispute of the professors of any science disparage the science itself,
Logics and Metaphysics are much more to be disparaged than Mathematics;
why, therefore, if I am half blind, must I take for my guide one who
can't see at all?

[p.\ 50]: So far as Mathematics do not tend to make men more sober and
rational thinkers, wiser and better persons, they are only to be
considered as an amusement, which ought not to take us off from serious


This tract may have had something to do with Bayes's election, in 1742,
to Fellowship of the Royal Society, for which the sponsors were Earl
Stanhope, Martin Folkes, James Burtow, Cromwell Mortimer, and John

William Whiston, Newton's successor in the Lucasian Chair at Cambridge,
who was expelled from the University for Arianism, notes in his memoirs
(p.\ 390) that `on August the 24th this year 1746, being Lord's Day, and
St.\ Bartholomew's Day, I breakfasted at Mr Bay's, a dissenting
Minister at Tunbridge Wells, and a Successor, though not immediate, to
Mr Humphrey Ditton, and like him a very good mathematician also'. 
Whiston goes on to relate what he said to Bayes, but gives no
indication that Bayes made reply.

According to Strange (1949) Bayes wished to retire from his ministry as
early as 1749, when he allowed a group of Independents to bring
ministers from London to take services in the chapel week by week,
except for Easter, 1750, when he refused his pulpit to one of these
preachers; and in 1752 he was succeeded in his ministry by the Rev.\
William Johnston, A.M., who inherited Bayes's valuable library.  Bayes
continued to live in Tunbridge Wells until his death on 17 April
1761\footnote{The \textit{Gentleman's Magazine} (see references below)
includes among the \textit{List of} \textsc{Deaths} \textit{for the}
Year 1760 ``Rev.\ Mr Bayes, at Tunbridge Wells'' on 7 rather than 17
April as given in this article.}.  His body was taken to be buried, with
that of his father, mother, brothers and sisters, in the Bayes and
Cotton family vault in Bunhill Fields, the Nonconformist burial ground
by Moorgate.  This cemetery also contains the grave of Bayes's friend,
the Unitarian Rev.\ Richard Price, author of the \textit{Northampton
Life Table} and object of Burke's oratory and invective in
\textit{Reflections of the French Revolution}, and the graves of John
Bunyan, Samuel Watts, Daniel Defoe, and many other famous men.

Bayes's will, executed on 12 December 1760, shows him to have been a man
of substance.  The bulk of his estate was divided among his brothers,
sisters, nephews and cousins, but he left \pounds200 equally between
`John Boyl late preacher at Newington and at Norwich, and Richard
Price now I suppose preacher at Newington Green'.  He also left `To
Sarah Jeffery, daughter of John Jeffery, living with her father at the
corner of Fountains Lane near Tonbridge Wells, \pounds500, and my watch
made by Elliott and all my linen and wearing apparell and household

Apart from the tracts already noted, and the celebrated Essay reproduced
here, Bayes wrote a letter on Asymptotic Series to John Canton,
published in the \textit{Philosophical Transactions of the Royal
Society} (1763, pp.\ 269--271).  His mathematical work, though small in
quantity, is of the very highest quality; both his tract on fluxions and
his paper on asymptotic series contain thoughts which did not receive as
clear an expression again until almost a century had elapsed.

Since copies of the volume in which Bayes's essay first appeared are not
rare, and copies of a photographic reprint issued by the Department of
Agriculture, Washington, D.C. U.S.A., are fairly widely dispersed, the
view has been taken that in preparing Bayes's paper for publication here
some editing is permissible.  In particular, the notation has been
modernized, some of the archaisms have been removed and what seem to be
obvious printer's errors have been corrected.  All the work of preparing
the text for the printer was most painstakingly and expertly carried out
by Mr M.\ Gilbert, B.Sc., A.R.C.S.  Thanks are also due to the Royal
Society for permission to reproduce the Essay in its present form.

In writing the biographical notes the present author has had the
friendly help of many persons, including especially Dr A.\ Fletcher and
Mr R.\ L.\ Plackett, of the University of Liverpool, Mr J.\ F.\ C.\
Willder, of the Department of Pathology, Guy's Hospital Medical School,
and Mr M.\ E.\ Ogborn, F.I.A., of the Equitable Life Assurance Society. 
He would also like to thank Sir Ronald Fisher, for some initial
prodding which set him moving, and Prof.\ E.\ S.\ Pearson, for patient
encouragement to see the matter through to completion.


\textsc{Anderson, J.\ G.}\ (1941). \textit{Mathematical Gazette},
\textbf{25}, 160--2.\newline
\textsc{Cantor, M.}\ (1908).  \textit{Geschichte der Mathematik}, vol.\
\textsc{iv}. (article by Netto.)\newline
\textsc{De Morgan, A.}\ (1960) \textit{Notes and Queries}, 7 Jan 1860,
  9--10, ``Queries: Rev.\ Thomas Bayes, etc.''\newline
\textsc{Loria, G.}\ (1933).  \textit{Storia delle Mathematiche}, vol.\
\textsc{iii}.  Turin.\newline
\textsc{Mackenzie, M.}\ (Ed.) (1865).  \textit{Imperial Dictionary of Universal
Biography}, 3 vols.\ Glasgow.\newline
\textsc{Strange, C.\ H.}\ (1949).  \textit{Nonconformity in Tunbridge
Wells}.  Tunbridge Wells.\newline
\textit{The Gentleman's Magazine} (1761 [April]), \textbf{31}, 188.\newline
\textit{Notes and Queries} [\textbf{180}] (1941), 19 April [282--283].


{\small[Since this biographical note was written, Mr O.\ B.\ Sheynin has
suggested that reference should be made to a second contribution from
Price, ``Supplement to the Essay on a Problem in the Doctrine of
Chances'' (\textit{Phil.\ Trans.}\ 1765, \textbf{54}, 296--335).  This
is concerned with improving approximations made in the main Essay. 


[From \textit{Biometrika} \textbf{45} (1958), 293--315; reprinted in
E.~S.~Pearson and M.~G.~Kendall (ed.), \textit{Studies in the History of
Statistics and Probability}, London: Griffin 1970.]