Who breaks a butterfly upon a wheel.

Some time ago as I was pulling Mr Pieret’s leg about the contents of his excellent blog, Thoughts in a Haystack, he countered by politely telling me to go away and write about an obscure mathematician, so today’s post about an obscure mathematician is dedicated to Mr Pieret because he, like the subject, is multi-talented, a good writer and is called John.

Today is the baptism day (not birthday which is unknown) of the Scots physician John Arbuthnot (1667 – 1735). Arbuthnot was a good friend of and a loyal lieutenant to Isaac Newton. He served in the Royal Society commission to kidnap John Flamsteed’s Star Catalogue and it was in fact Arbuthnot who carried out the dirty deed, travelling to Greenwich to collect the manuscript. He also served in the Royal Society commission called into being to investigate Leibniz’ complaints concerning the origins of the calculus. A commission that directed by Newton from the shadows found against Leibniz and for Newton. The friendship between Newton and Arbuthnot was slightly strange as Arbuthnot was a politically active High Church Tory and Newton was a Unitarian and politically active Whig in a time when the boundaries between the political and religious parties in England were extreme.


John Arbuthnot, by Godfrey Kneller Source: Wikimedia Commons

Arbuthnot was a mathematician whose claim to fame was based on his translation into English of Christian Huygens’ book on probability theory (the first English text published on probability) and on his statistical analysis of the male and female birth rates in England. This is probably the first use of probability in a social statistical analysis and the earliest case of a statistical significance test. Arbuthnot also published a highly praised essay on the usefulness of mathematics, which was not a generally accepted concept in England at the time, for as Arbuthnot pointed out mathematics was not on the syllabus of a single English grammar school. A passionate numismatist he also contributed to the development of archaeology and history with his papers on Tables of Grecian, Roman, and Jewish measures, weights and coins; reduced to the English standard.

From profession Arbuthnot was a physician and was for many year personal physician to Queen Anne. However mathematics and medicine were by no means the limits of his intellectual activities. He was a close friend of the composer Handel and even wrote librettos for him. His musical interests led to his appointment as director of the Royal Academy of Music from its inception in 1719 until 1729. However within the history of English culture he is best known as a literary satirist. A friend and literary colleague of Swift, Pope and Gay he is the creator of the legendary caricature of the English man, John Bull. Together with the others he was a member of the Scriblerus Club a short-lived satirical collective. The title of this post is a famous quote from Pope’s Epistle to Dr. Arbuthnot. For modern politicians his most useful piece of writing would be his Treatise of the art of political lying.

Arbuthnot is a fascinating polymath who today is only known to historians of English literature, even those who use the image of John Bull as the archetypal Englishman have no idea who created him, and historians of statistics. However he was a small but significant member of the scientific community at the beginning of the 18th century of the type without whom science cannot develop.


Filed under History of science, Newton

13 responses to “Who breaks a butterfly upon a wheel.

  1. DonE

    Thanks, Thony, I enjoyed this one, particularly the conclusion “…of the type without whom science cannot develop”. Arbuthnot definitely sounds like he had an interesting life story, but wasn’t the man in front. But science as a social process is only so modern– there have always been people like Arbuthnot who stand in the community of discovery without much historical mention. Cheers to the internet for bringing some of them some latter-day exposure!

  2. Once again I advance historical understanding!

    You may be interested in this on Arbuthnot from Elliott Sober’s Evolution and Evolution: The Logic Behind the Science:

    One landmark in the development of probability theory during this period was a version of the design argument published by John Arbuthnot, who was physician to Queen Anne and inventor of the satirical character John Bull. Arbuthnot’s “Argument for Divine Providence, Taken from the Constant Regularity Observ’d in the Births of Both Sexes” appeared in the Philosophical Transactions of the Royal Society for 1710. The paper provides a tabulation of eighty-two years of London christening records; more boys than girls are listed for each year. Arbuthnot takes this difference at face value; he must have realized that not every birth was recorded, but he nonetheless assumes that the records reflect a real difference in the frequencies of male and female births. The main part of the paper is given over to the task of calculating the probability that this pattern would obtain if the sex ratio were due to chance. By “chance” Arbuthnot means that each birth has a probability of 1/2 of being a boy and 1/2 of being a girl. According to this hypothesis, there being more boys than girls in a given year has the same probability as there being more girls than boys in that year; the chance hypothesis also allows for a third possibility, namely, there being exactly as many girls as boys:

    Pr (more boys than girls born in a given year | Chance)
    = Pr (more girls than boys born in a given year | Chance)
    >> Pr (exactly as many boys as girls born in a given year | Chance) = e.

    Although Arbuthnot goes to the trouble of explaining how e might be calculated, the details of his calculation don’t matter to the argument; the point is just that for each of the years surveyed, e is tiny. Arbuthnot concludes that the probability of there being more boys than girls in a given year, according to the chance hypothesis, is just under 1/2, and so the probability of there being more boys than girls in each of eighty-two years is less than (1/2)^82. He further asserts that if we were to tabulate births in other years and other cities, we would find the same male bias. So, the probability of all these data — both the data that Arbuthnot presents and the data that he does not have but speculates about — is “near an infinitely small quantity, at least less than any assignable fraction.” The conclusion is obvious: “it is Art, not Chance, that governs.”

    Arbuthnot also notes that males have a higher mortality rate than females, so that the male bias at birth gradually gives way to an even sex ratio at the age of mariage “We-must observe,” he says,

    that the external accidents to which males are subject (who must seek their food with danger) do make a great havock of them, and that this loss exceeds far that of the other sex, occasioned by diseases incident to it, as experience convinces us. To repair that loss, provident Nature, by the disposal of its wise creator, brings forth more males than females.

    At the end of the paper, Arbuthnot adds, as a scholium, that

    polygamy is contrary to the law of nature and justice, and to the propagation of the human race. For where males and females are in equal number, if one man takes twenty wives, nineteen men must live in celibacy, which is repugnant to the design of nature, nor is it probable that twenty women will be so well impregnated by one man as by twenty.

    In Arbuthnot’s hands, the design argument begins as an explanation of what is, but ends as an argument concerning what ought to be.

    • DonE

      I guess that would also make sense for a satirist with such a diverse range of publications and interests. I mean, at the heart of it satire is too about what is and what the author thinks should (usually should not) be.

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  4. Ian H Spedding FCD

    Thank you for this and for the preceding piece on the sugar trade. You have the knack of making these subjects fascinating. If I had had someone like you teaching history when I was at school I would definitely have paid more attention.

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