J.D. Forbes and the development of curve plotting


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Belatedly, I have uploaded the slides from my presentation at the MAA Mathfest last year, on J.D. Forbes and the developent of curve plotting. Here’s the basic idea.

When, in 2012, experimental evidence for the Higgs boson was announced, it came in the form of a curve with a blip, immediately understood by the audience. Yet 190 years earlier, in 1823, the practice of curve plotting was so unusual that S. H. Christie felt it necessary to explain not only the meaning of the curve for magnetic variation that he presented in the Philosophical Transactions but also the process of defining the axes, representing the data as dots, and drawing the curve.

The development of curve plotting as a technique for relating observational data to mathematized theory appears to have been surprisingly difficult. Early promoters, such as Lambert, were not followed, and not until the 1830s did the method start to spread, following the work of Playfair and Quetelet in statistics, and Herschel and Forbes in natural philosophy (Beniger & Robyn 1978; Hankins, 1999; Tilling 1975).

Tilling identifies a step change in the ubiquity of curve plotting among scientists, initiated by J.D. Forbes, Professor of Natural Philosophy at Edinburgh 1833-1859. Beginning in 1834, he used curves both to present and to analyse observational results relating to heat, meteorology, and glacial flow. This was really handy for me, as Forbes later  became Principal of the United College of St Andrews University (effectively Principle of the whole university), and all his archives are here on my doorstep.

Based on an investigation of Forbes’ notebooks, the paper looks in particular at his curve for daily oscillations in atmospheric pressure, to analyse the the influences on his use of curves. I tentatively conclude that:

  • The form of the horary oscillation curve is a response to very particular circumstances of Forbes’ life in 1831-3. In most other work he used  curves as labour saving and calculational tools
  • Transactions of the RSE encapsulates a strongly visual culture in Scottish science in the 1820s-30s
  • The implicit assumption of an analogue space may have been a barrier to the adoption of curve plotting techniques until the 1830s when increasing confidence in instruments and new ways of seeing allowed their acceptance by authors and audiences


Beniger, James R., and Dorothy L. Robyn. ‘Quantitative Graphics in Statistics: A Brief History’. The American Statistician 32, no. 1 (1 February 1978): 1–11. doi:10.2307/2683467.

Hankins, Thomas L. ‘Blood, Dirt, and Nomograms: A Particular History of Graphs’. Isis 90, no. 1 (1 March 1999): 50–80.

Tilling, Laura. ‘Early Experimental Graphs’. The British Journal for the History of Science 8, no. 03 (1975): 193–213. doi:10.1017/S0007087400014229.


Entry of ‘mechanism of the heavens’ into English


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You’ll have gathered that I like playing around with google’s Ngram viewer.

Having been thinking about Mary Somerville recently, I thought I’d use it to assess in some sense the impact of her first book Mechanism of the Heavens, an interpretation of Laplace‘s Mécanique Céleste for the British (published 1832). The results are intriguing. Because the Ngram viewer is case sensitive, one can distinguish between deliberate references to the book, and occasions when the phrase is being used as part of the language (slightly complicated by being in an age when writers still capitalised important words even when they were not titles). But, basically, the phrase was bumbling along at a very low incidence in the English language until the run up to the publication of the book, when usage suddenly increases dramatically. It looks as though Somerville was largely responsible for popularising the phrase in English, as well as for explaining Laplace.


Mary Somerville and John Herschel


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Mary Somerville, in her reminiscences, gives the impression (though she does not say definitely) that she first met John Herschel when she visited his father, William Herschel, in 1816.

However, a letter I came across in the summer in the Herschel papers at St John’s College in Cambridge, shows that they had actually met in 1812 – though he did not mention her by name, and he doesn’t seem to have made an impression on her. But the dates correspond to the Somervilles’ visit to London soon after they were married, and it seems unlikely that there were two such ladies around at the same time.

In August 1812, John Herschel wrote to his friend Rev. John William Whittaker :

I had the happiness of an introduction to a Dulcinea who reads the Mec. Cel__ and Mec. Analytique _ What say you to that?_ To use Playfair’s expressions she “possesses a degree of Math. Knowledge rarely found in the other sex and is at once an ornament & example to her own – to her Country & to the human race!! _ No mortal however would have been able to read a page of La Grange in her eyes. Or to discover the Analyst in her manner, her conversation or her appearance  NB. Corol. She is Confoundedly not-beautiful. Of course one ought never positively to affirm the reverse of beauty to belong to any woman_  (St John’s College, Herschel papers, Herschel/box1/letter 1)

His views on whether female mathematicians can or should be beautiful are intriguing and well worth exploring, and play very well to the themes that Eva Kaufholz explored in her talk on Sofja Kowalewskaja at the Mathematical Biography conference last month.

No Actual Measurement was required…


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I’ve just put a preprint of my article, ‘No actual measurement … was required: Maxwell and Cavendish’s null method for the inverse square law of electrostatics’ on the Arxiv. It has grown and developed from my conference presentation last year – with thanks for great comments and stimulating discussions with Daniel Mitchell and the referees. It’s been accepted (hooray!), but still some minor corrections before the final published version.

Experimentalists in Maxwell’s Cavendish


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I was very excited recently to discover that a colleague was descended from Charles Heycock, who more or less founded Materials Science in Cambridge. However, Heycock’s significance for me is that he entered, and published from, Maxwell’s Cavendish Laboratory aged 17 while still at school. He exemplifies Maxwell’s open door policy at the Cavendish (provided you were male).

This prompted me to post up an analysis of those we know were doing experimental work in Maxwell’s Cavendish, plotted against their year(s) pre or post graduation and their subject. The results are remarkable and show very clearly the impact of the Cambridge system on experimenter numbers.


On the chart, zero represents their graduation year. Thus, -1, -2… etc are undergraduates at one, two, … etc years before graduation. 1, 2, … etc are postgraduates at one, two, … etc years after graduation.

The majority of postgraduates were mathematicians. They were hanging around Cambridge seeking research topics that would earn them a College fellowship in a year or two’s time. There was no such incentive for Natural Science postgraduates. In 1870 the University Reporter commented that some colleges would consider themselves ‘guilty of extravagance’ in appointing scientific fellows.

Conversely, the undergraduates were mostly studying Natural Sciences. From 1874 on a practical exam in physics had been included in the Natural Sciences course, providing an incentive for students to take an interest in experimental physics – though it is important to note that the students included in the chart here were doing research not a practical physics course. No mathematicians are recorded as doing experimental work while undergraduates. Although physical subjects had been added into the Maths Tripos in 1873, students very rapidly realised that there was so much choice in the question paper that they had no need to study physics in order to do well. Nor did they need practical work. Physics was removed again in 1882 to the postgraduate Part III of the Maths Tripos.

Thus the chart echoes George Bettany’s lament to Nature in 1874,

‘The great hindrance to the success of the Cavendish Laboratory at present is the system fostered by the Mathematical Tripos. The men who would most naturally be the practical workers in the laboratory are compelled to refrain from practical work if they would gain the best possible place in the Tripos list. Very few have courage so far to peril their place or to resign their hopes as to spend any valuable portion of their time on practical work… For a man to do practical work in physics at Cambridge implies considerable exercise of courage and self-sacrifice.’

And what of Charles Heycock? He is the “non-Cambridge” undergraduate at <-4. He was 17 when he worked in the Cavendish with Arthur Clayden (an undergraduate) on the spectrum of indium. He didn’t enter the University for another year, and subsequently read Natural Sciences, graduating with a first class degree in chemistry and physics. He became a FRS and founder of Cambridge’s Department of Materials Science.

No actual measurement was required


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I’ve just put up a copy of my presentation on Cavendish, Maxwell, and the inverse square law of electrostatic repulsion, given at the Making of Measurement Conference in Cambridge last week.

This poses the question of what Maxwell and his student Donald MacAlister, get out of repeating Cavendish’s null experiment on the inverse square law.  The abstract read,

Traditionally, the foundation of the theory of electrostatics has been taken to be Coulomb’s 1785 torsion balance experiments, reified as “Coulomb’s Law”. However, Coulomb’s results, and interpretation, were frequently challenged, notably by Volta and Simon and, as late as 1836, by William Snow Harris.

In the first edition of the Treatise on Electricity and Magnetism (1873), Maxwell acknowledges Coulomb’s experiments as establishing the inverse square law, merely to dismiss them again as demonstrating it only to a rough approximation. Instead he cites the observation that a charged body, touched to the inside of a conducting vessel, transfers all its charge to the outside surface of the vessel, as ‘far more conclusive than any measurements of electrical forces can be’ (#74). This assertion was based on mathematical proof that an exact inverse square law was a necessary condition for electricity to rest in equilibrium on the surface of a conductor.

The following year Maxwell acquired the hitherto unpublished electrical researches of Henry Cavendish, and found that around 1771 Cavendish had conducted a (fairly) rigorous test of the mathematically predicted null result concluding that the negative exponent in the force law could not differ from 2 by more than about 1/50. Maxwell and a research student, Donald McAlister, created their own version of Cavendish’s experiment, achieving a claimed sensitivity of 1/21600. By the second edition of the Treatise (1881) his previous, ‘…far more conclusive than any measurements…’ has become ‘… a far more accurate verification of the law of force [than Coulomb’s]’ (#74). In his draft for the Cambridge Philosophical Society, Maxwell wrote, ‘Cavendish thus established the law of electrical repulsion by an experiment in which the thing to be observed was the absence of charge on an insulated conductor. No actual measurement of force was required. No better method of testing the accuracy of the received law of force has ever been devised’ (my emphasis).

 More recently, Dorling (1974) has explored the sense in which it was rational for Cavendish and Maxwell to generate an entire law from a single (null) data point, while Laymon (1994) has pointed out the circularity of Maxwell’s argument and located the actual measurement in the testing of the sensitivity of the electrometer.

 Taking Laymon’s and Dorling’s critiques on board, and drawing on works and papers by Maxwell, Kelvin, Tait and Harris, this paper will examine how Maxwell and his contemporaries understood what Cavendish had done, what they thought the null method achieved, and the value to them of recreating the experiment.


Dorling, Jon. ‘Henry Cavendish’s Deduction of the Electrostatic Inverse Square Law from the Result of a Single Experiment’. Studies in History and Philosophy of Science Part A 4, (1974): 327–48.

Laymon, Ronald. ‘Demonstrative Induction, Old and New Evidence and the Accuracy of the Electrostatic Inverse Square Law’. Synthese 99, (1994): 23–58.

Editing Cavendish: Maxwell and The Electrical Researches of Henry Cavendish


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I’ve recently put the preprint on the arXiv of my conference presentation last September, on Maxwell’s edition of The Electrical Researches of Henry Cavendish.

Although, probably, Maxwell’s least significant contribution to science, I argue that we can learn a lot about his attitudes and preoccupations during the last five years of his life from the way he tackled this task.

Is crowdsourcing good for us?


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Last week I was listening in on a fascinating conversation between a group of research mathematicians about Polymath.  Polymath is a community of massively collaborative online mathematical projects which is achieving impressive results and is widely cited as an exemplar of the benefits of crowd sourcing.

Yet despite its apparent openness, my group felt excluded from Polymath – excluded by the pace of results production. There are, apparently, only a handful of mathematicians in the world clever enough to contribute effectively and the rest are left gasping behind. My group argued explicitly that what is important about mathematics is not getting results, but being engaged in the process, and Polymath does not allow this in practice even though it might in theory.

Their comments reminded me of James Clerk Maxwell’s remark to Arthur Schuster that, ‘The question whether a piece of work is worth publishing or not depends on the ratio of the ingenuity displayed in the work to the total ingenuity of the author.[1]. Like the mathematicians, he was arguing that engagement was the important thing, not results. For Maxwell there was an intensely moral imperative behind this view. Engagement in an abstract cause such as maths or physics helped one control baser desires. I’m not sure my mathematicians would have gone this far, but the value to society of participation is well worth thinking about.

Maxwell, of course, was one of the lucky ones. He was one of the handful that would have been able to keep up. And in fact he indulged in collaborative exchanges with William Thomson and Peter Guthrie Tait that bear many of the hallmarks of Polymath – all conducted on postcards through the Royal Mail [2]. These collaborations were thus less visible than Polymath. Were they less discouraging to contemporaries who might be engaging in their own collaborations at their own levels?

It seems that there is not much new in collaborative mathematics, but that being online and very visible may have downsides as well as benefits.

[1] Schuster, A. (1910) in A history of the Cavendish laboratory, 1871-1910, (Longmans, Green, London, 1910) p.32

[2] P. M. Harman ed. (1995-2002) The Scientific Letters and Papers of James Clerk Maxwell, vols I, II, III Cambridge University Press.

Highlights of the OER4Adults SWOT survey | oer4adults.org


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Highlights of the OER4Adults SWOT survey | oer4adults.org. These are the draft highlights of the OER4Adults project I have been working on, together with Lou McGill, Allison Littlejohn, and Eleni Boursinou, for the past 8 months. OER4Adults takes an overarching view of Open Educational Resources in adult and lifelong learning across Europe. Still a lot more analysis we could do, but the initial results are interesting. We conclude that open educational resources, and the practices associated with these resources, are an immensely powerful idea that potentially make a significant difference to education systems, but under-estimating the degree of cultural change needed to optimise their value, and the power of vested interests endangers realising this potential and the visions for lifelong learning 2030

New light on J J Thomson’s appointment as Cavendish Professor


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J J Thomson, nobel prize-winner and widely credited with discovering the electron, caused a degree of consternation in Cambridge in 1884 when he was appointed to succeed Lord Rayleigh as Professor of Experimental Physics at the Cavendish Laboratory in Cambridge. Thomson was only 28 and known more as a theoretical than an experimental physicist; he was not the obvious candidate.

I have recently found evidence showing that before putting himself forward, Thomson actually spearheaded a campaign to attract another candidate to the post. This candidate was Sir William Thomson (no relation to J J), the most eminent physicist of the day.  Sir William had been asked to fill this post twice before and had refused each time: once when the Laboratory was founded in 1871 ( Clerk Maxwell was appointed instead), and again in 1879 on Maxwell’s death. 

The Cambridge University Registry Guard Book for physics (a sort of scrapbook archive of documents relating to the physics laboratory) contains a copy of a memorial addressed to Sir William, urging him to take the chair. The campaign was led by J J and all members of the university who supported it were urged to contact him[1].

The case may have seemed hopeless, and J J had, so far, managed significantly fewer signatures on his memorial (24) than the equivalent petition to Lord Rayleigh five years earlier in 1879 (around 100) [2]. But it is interesting in showing that J J was already asserting his leadership within the Cambridge physics community – and raising his profile in this way clearly did him no harm when he came to apply for the post himself.

[1] Cambridge University Library, University Registry Guard Book, CUR 39.33, item 66

[2] Cambridge University Library, University Registry Guard Book, CUR 39.33, item 56