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“But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton UK Séminaire d'Histoire du Calcul des Probabilités et de la Statistique Paris February 2 nd 2007 1

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Page 1: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

“But You have to Remember P. J. Daniell of Sheffield”

or Percy Daniell and the British

Probability Tradition(s)

John AldrichUniversity of Southampton UK

Séminaire d'Histoire du Calcul des Probabilités et de la Statistique Paris February 2nd 2007

1

Page 2: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

David Kendall on the origins of a British probability tradition

But you have to remember P. J. Daniell of Sheffield. Daniell wrote his major papers in the US in the South—I

think. Who taught him? Sheffield does not have a portrait. When he went to Sheffield he apparently gave up

probability and started working on the design of blast furnaces.

2

Page 3: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Aims of this talk

Mainly to get the basic facts about Daniell straight

and to dispel some of the mystery

but also to show why Daniell should be considered in

the search for a British probability tradition to discuss whether he belongs …

Page 4: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Why consider him?

In books on measure & probability

Daniell is often the only 20th century British mathematician to appear.

With de Moivre and Bayes he is often the only British mathematician to appear.

Page 5: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Daniell and probability

“Major papers” Daniell integral 1918Daniell Kolmogorov Extension theorem 1919

Two unnoticed papersOrder statistics 1920Dynamic probability 1921

Time series contributionDaniell window 1946

Page 6: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Percy John Daniell (1889-1946) in his 20s?

Page 7: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Main activities

Theoretical physicist

Pure mathematician

Control theorist

Applied mathematician 7

Page 8: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Percy John Daniell (1889-1946) Born 9th of January 1889 in Valparaiso Chile.

First child of William and Florence.

Family returned to Birmingham in 1895.

William's occupation: export merchant's buyer.

Page 9: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Places: education & employment King Edward’s 1900-1907 Trinity Cambridge 1907-1911 Liverpool 1911-1912 Göttingen 1912-1913 Rice Institute Houston 1913-1923 Sheffield 1923-1946

Page 10: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

King Edward’s School “one of the great schools of England”

ty tz can be

Page 11: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

King Edward’s and the Cambridge connection

The school was very successful at getting its pupils into Cambridge

Once there they did very well.

Page 12: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

“The study of Higher Mathematics in the British Empire is now practically concentrated at Cambridge”

The Times Trinity College: biggest and best

Page 13: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Mathematics Tripos Part I

The first class students wranglers were placed in order of their marks

Senior Wrangler, Second Wrangler, Third Wrangler, etc.

The order was published Students labelled for life—e.g. Keynes

forever 11th Wrangler

Page 14: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

MT Part I: retrospectives

J. E. Littlewood (1905, Senior Wrangler) I wasted my time except for rare interludes. One had to spend two-thirds of the time practising how to solve

difficult problems against time. The game we were playing came easily to me and I even felt a

satisfaction of a sort in successful craftmanship.

H. Hassé (1905, 7th Wrangler) the great absence from the teaching—what we now know as

analysis. The real mathematician ... will survive the effects of any teaching

and of any syllabus.

Page 15: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Daniell Last Senior Wrangler “Killing a tradition”

Page 16: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Switch to Physics: Part II of the Natural Sciences Tripos Teachers: J J Thomson and …

First class degree

Rayleigh Prize “Diffraction of light for the case of a hole in a plane of perfectly reflecting screen.”

Page 17: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Assistant Lecturer in Liverpool 1911-2: high wrangler’s rite of passageW. H. Young (1863-1942) part-time lecturer.

Possible influence?

Page 18: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

To the new Rice Institute as Assistant Professor in Applied Mathematics. Sponsor J. J. Thomson.

Page 19: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Rice Institute—after a few years

Page 20: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

But first to Göttingen for a year on a $1000 travelling fellowship from Rice “studied under Born and Hilbert”

published a paper on relativity with Ludwig Föppl (one of Hilbert’s physics PhDs)

“On the Kinematics of the Born Rigid Body”

Page 21: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Then to Houston

Houston not much more than an overgrown commercial village seventeen hundred miles away from the American metropolis

It is hard for an Englishman to realise that the civilisation of the whole of the area west of the Mississippi ... is to all intents and purposes the product of the fifty short years since the civil war

Julian Huxley "Texas and Academe"

Page 22: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Main colleague Griffith Evans (1887-1973) ass. pro. pure mathematics, integral equations & functionals. Sponsor Volterra.

Page 23: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Daniell’s research at Rice

Applied maths/ theoretical physics in 1915 The Coefficient of End-correction (out of

Rayleigh and Cambridge)

Rotation of Elastic Bodies and the Principle of Relativity (out of Love and Born)

Page 24: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

And then pure mathematics…..

1918 A General Form of Integral (Daniell integral)

1919 Integrals in an Infinite Number of Dimensions (Daniell-Kolmogorov extension theorem)

Etc., etc. …

Page 25: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

A General Form of Integral: wonderful prospectus [Earlier theories of integration] are based on

the fundamental properties of sets of points in a space of a finite number of dimensions. In this paper a theory is developed which is independent of the nature of the elements.

They may be points in a space of a denumerable number of dimensions or curves in general or classes of events so far as the theory is concerned.

Page 26: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Integrals in an Infinite Number of Dimensions

No probability application envisaged

Daniell did not follow this up…

Page 27: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Where did this work come from?

Analysis—out of Young, Radon …

Strange that Daniell did this work but not strange that somebody did it and did it then.

Related work by Evans and a PhD, Hubert Bray.

Page 28: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Unnoticed paper I “Observations Weighted According to Order” 1920 “at least thirty years ahead of its time, for it took that

long for his major results to be rediscovered.” Stigler.

the ordinary average or mean, the median, the discard-average … can all be regarded as calculated by a process in which the measures are multiplied by factors which are function of order.

It is the general purpose of this paper to obtain a formula for the mean square deviation of any such expression. The formula may be used to measure the relative accuracies of all such expressions.

Page 29: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Unnoticed paper II “Integral Products and Probability” 1921 In many problems arising in statistical biology and

statistical economics time enters as an indispensable factor.

the aim of this paper is to provide a form of analysis suitable for such problems .... The first step in the analysis is a search for some standard formula on which may be built a more complex and general theory.

It is found that … a functional equation is satisfied which is expressed in terms of a Stieltjes integral product. ... The Stieltjes integral product itself forms a second nucleus for our paper

Page 30: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Where did it go ? Norbert Wiener (1894-1964) 1919 Wiener appointed instructor at MIT.

1920-22 Wiener writes 4 papers using the Daniell integral—3 on Brownian motion.1922 and -28 Wiener used Daniell as a referee for jobs.

Page 31: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Daniell & Wiener

They probably met at the International Mathematical Congress in Strasbourg 1920.

They do not refer to each other’s work in probability

Their work was very different in intended application and in technique.

Page 32: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Being a prof in England: Daniell to Wiener 1922 I wrote to London university my opinion of your suitability for the

position and it was partly praise and partly otherwise. That is to say I think highly of your promise as a mathematician but—and, I naturally expect you to disagree with me on this—I feel that you have not yet attained as established a position & have not had as much experience as they try to get for such a position.

A professor in England is not merely a man of professorial rank &

ability—he is a very active & sometimes autocratic head of a department.

I doubt it would be good for you if your application were successful, You would be rather loaded down with work and would most likely get stuck in a rut.

Page 33: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Daniell—the 20s and return to England 1922 Cambridge ScD

1923 To Sheffield as Professor of Mathematics

Analysis research continues but with less intensity through the 20s

Page 34: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Sheffield The city being an important centre of steel, electro-plate, glass, and other manufacturing industries, and in the heart of an extensive coal-mining area, students of Engineering, Metallurgy, Mining, Fuel Technology and Glass technology have exceptional opportunities.

Page 35: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

On being a prof continued

In 1928 Wiener applied for a chair in Melbourne. Daniell told Wiener

It's quite time I did some work myself but a Chair in England involves a great deal of business which is done in America by the office.

Page 36: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

The 1930s

One publication

The Theory of Flame Motion, Proceedings of the Royal Society of London, A, 1930

Related to work for Safety in Mines Research Board in Sheffield.

Page 37: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

According to Stewart (obit.)

Daniell was a prodigious reader of scientific journals and was conversant with the latest developments in Physics, Chemistry and Biology as well as those in most branches of pure mathematics.

he seldom gave his undivided attention to the systematic development of particular lines of research and therefore the actual number of his publications does not provide a true measure of his intellectual powers of and activity.

Much of his time and energy was expended in advising and assisting research workers in many fields and it was only on rare occasions that he troubled to make a permanent record of his own contributions to the problems involved.

Page 38: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Stewart continued

Lots of university committee work Outside interests included the training of

teachers, the Mathematical Association, and to the School Certificate Examination.

In one of the 5 letters Daniell writes to Mordell saying he can’t make a meeting with Hadamard because of press of school exam business.

Page 39: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton
Page 40: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Activities in the WarTeaching continued—intensified in fact.

7 classified papers Mainly on fire-control Daniell worked with Arnold Tustin and Arthur

Porter important figures in automatic control after the war

Daniell translated American work by Bode and Wiener so that British engineers could understand it.

Page 41: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Arnold Tustin and the describing function The general method is based on unpublished

work of the late Professor P. J. Daniell, who provided an analytical treatment of the effect of backlash of which the present paper is essentially an interpretation in geometrical terms.

Journal of the IEEE 1947

Page 42: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Porter recalled in 1965

Daniell's subsequent contributions to servo theory, although not widely known because his reports and memoranda were security classified, were of high significance.

Daniell was the first man in Europe to ‘translate’ Norbert Wiener's work on the interpolation and extrapolation of stationary time series

Daniell's interpretations of the early Wiener papers on control theory are refreshingly elegant and make a noteworthy contribution to the evolution of control-systems engineering in Britain.

Page 43: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Illness and death. Stewart (obit.)

The strain of the war years became evident during the summer of 1945 when he was attacked by serious heart trouble.

He recovered to some extent and decided to undertake the work of the session 1945-1946, but there seems little doubt that his life would have been prolonged if he had made a different decision.

He continued with his many activities in a spirit of great fortitude and determination, but early in May, 1946, he collapsed at his home and died a few weeks later without fully recovering consciousness.

Page 44: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Daniell and the British probability tradition(s)

The “major papers” were canonised decades after they were written and years after his death.

There was, though, one immediate canonisation, commemorated in the eponym, the "Daniell window."

Page 45: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

“Daniell window” RSS Symposium January 1946. Daniell’s contribution begins My absence from this symposium is a grief to

me. The work done in America has been based

on a fundamental study by N. Wiener of integrals in an infinite number of dimensions, corresponding to the values of the fluctuating quantity at various instants. The work is not behind that of the Russian school in time or importance.

Page 46: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Maurice Bartlett (left) introduced Daniell into the time series tradition

Page 47: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Probability/measure comes to Britain and Daniell becomes known in the 50s/60s.

Lévy (1925) knew Wiener’s work and through it Daniell (1918).

Doob (1934, 1953) refers to Daniell’s 1919 paper.

Stone developed Daniell (1918)

Page 48: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Daniell’s Style Daniell did pure maths, applied maths and physics

and wrote in the existing style of those subjects.By contrast Wiener mixed up the styles “I saw as my habit, a physical and even an

engineering application, and my sense of this often determined the images I formed and the tools by which I sought to solve my problems...”

“Paley and I attacked the problem of the conditions restricting the Fourier transform of a function vanishing on the half line. This is a sound mathematical problem on its own merits, and Paley attacked it with vigor, but what helped me and did not help Paley was that it is a essentially a problem in electrical engineering.”

Page 49: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Puzzles

Why did he switch to pure mathematics in 1915-18?

Why did he stop writing when he went to Sheffield?

Page 50: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

As a person …Daniell married Nancy Hartshorne in 1914. They had two daughters,

Frances and Mary, and two sons, David and John.

Daniell impressed all who came into contact with him by his great integrity of character and his sincerity of purpose.

He disliked publicity and his tastes were simple. He delighted in good music, in books, in friendly discussion, in country walks and in the quiet pleasures of a happy family life.

Stewart 1947

I have very fond memories of him. He was charming, delightful, low key, modest.

Arthur Porter phone conversation 20th October 2006

Page 51: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

Works by Daniell: about 50 papers (Starred items not in Stewart’s list.)

*P. J. Daniell (1912) "Diffraction of light for the case of a hole in a plane of perfectly reflecting screen." Rayleigh Prize Essay. P. J. Daniell (1915) The Coefficient of End-correction I, Philosophical Magazine, 30, 137-146. P. J. Daniell (1915) The Coefficient of End-correction II, Philosophical Magazine, 30, 248-256. P. J. Daniell (1915) Rotation of Elastic Bodies and the Principle of Relativity, Philosophical Magazine, 30, 756-761. *P. J. Daniell (1917) Translation of E. Borel's 1911 Inaugural Address "Monogenic Uniform Non-analytic Functions," Rice Institute Pamphlet, 4, No. 1, P. J. Daniell (1917) New Rules of Quadrature, American Mathematical Monthly, 24, 109-112. P. J. Daniell (1917) The Modular Difference of Classes, Bulletin of the American Mathematical Society, 23 446-450. P. J. Daniell (1918) A General Form of Integral, Annals of Mathematics, 19, 279-294. P. J. Daniell (1918) Differentiation with Respect to a Function of Limited Variation, Transactions of the American Mathematical Society, 19, 353-362. P. J. Daniell (1918) Integrals around General Boundaries, Bulletin of the American Mathematical Society, 25, 65-68. P. J. Daniell (1919) A General Form of Green's Theorem, Bulletin of the American Mathematical Society, 25, 353-357. P. J. Daniell (1919) The Derivative of a Functional, Bulletin of the American Mathematical Society, 25, 414-416. P. J. Daniell (1919) Integrals in an Infinite Number of Dimensions, Annals of Mathematics, 20, 281-288. P. J. Daniell (1919) Functions of Limited Variation in an Infinite Number of Dimensions, Annals of Mathematics, 21, 30-38. *P. J. Daniell (1919) 2738: Solution to a Problem posed by W. D. Cairns, American Mathematical Monthly, 26, 321. P. J. Daniell (1920) Further Properties of the General Integral, Annals of Mathematics, 21, 203-220. P. J. Daniell (1920) Stieltjes Derivatives, Bulletin of the American Mathematical Society, 26, 444-448. P. J. Daniell (1920) Stieltjes-Volterra Products, Comptes Rendus du Congrès International des Mathématiciens, 22-30 Septembre 1920 / Publiés par Henry Villat . 1921 P. J. Daniell (1920) Observations Weighted According to Order, American Journal of Mathematics, 42, 222-236. *P. J. Daniell (1921) The Integral and its Generalizations, The Rice Institute Pamphlet, Volume Eight, No. 1. pp. 34-62. P. J. Daniell (1921) Integral Products and Probability, American Journal of Mathematics, 43, 143-162. P. J. Daniell (1921) Two Generalizations of the Stieltjes Integral, Annals of Mathematics, 23, 168--182. *P. J. Daniell (1923) Review of An Introduction to Electrodynamics by Leigh Page. 39. P. J. Daniell (1924) The Setting of a Proposition, Annals of Mathematics, 26, 65-78. P. J. Daniell (1926) Derivatives of a General Mass, Proceedings of the London Mathematical Society, 26, 95-118. P. J. Daniell (1926) Disussion of "Theory of Mine Ventilation," Transactions of the Institute of Mining Engineers, 71, 39-45. P. J. Daniell (1926) Orthogonal Potentials, Philosophical Magazine, 7, 247-258. P. J. Daniell (1927) A Note on Schrödinger's Wave Mechanics, Journal of the London Mathematical Society, 2, 106-108. P. J. Daniell (1928) Transformations of Limited Variation, Proceedings of the London Mathematical Society, 29, 537-555. P. J. Daniell (1928) Stieltjes Derivatives, Proceedings of the London Mathematical Society, 30, 187-192. P. J. Daniell (1929) Boundary Conditions for Correlation Coeffcients, British Journal of Psychology, 20, 190-194. P. J. Daniell (1930) The Theory of Flame Motion, Proceedings of the Royal Society of London, A, 126, 393-405. P. J. Daniell (1940) Ratio Tests for Double Power Series, Quarterly Journal, 2, 183-192. P. J. Daniell (1940) Remainders in Quadrature and Interpolation Formulae, Mathematical Gazete, 24, 238-244. *P. J. Daniell (1942) Analogy between the Interdependence of Phase-shift and Gain in a Network and the Interdependence of Current and Potential Flow in a Conducting Sheet, Report in Servo Panel Library B. 39. *P. J. Daniell (1943) Interpretation and Use of Harmonic Response Diagrams (Nyquist Diagrams) with Particular reference to Servomechanisms, Report No. 1 and pp. 1-12 of Selected Government Research Reports Volume 5: Servomechnisms, London, Her Majesty's Stationery Office 1951. *P. J. Daniell (1944) Operational Methods for Servo Systems, Servo Panel Report S1, July 1944 published as Report No. 2 and pp. 13-33 of Selected Government Research Reports Volume 5: Servomechnisms, London, Her Majesty's Stationery Office 1951. *P. J. Daniell (1944) Digest of Manual on the Extrapolation, Interpolation and Smoothing of Stationary Time Series with Engineering Applications, by Norbert Wiener, OSRD Report 370, Servo Panel Library, p. 47, circa 1944. *P. J. Daniell (1944) Backlash in Reset Mechanisms, C. S. Memo 199, 16 March 1945. *P. J. Daniell (1944) An Explanatory Note on H. W. Bode's Paper on the Relation between Phase-lag and Attenuation (Bell Journal 19 (1940) p. 421), C. S. Memo 201, 21 March 1945. *P. J. Daniell (1946) Contribution to Discussion in the Symposium on Autocorrelation in Time Series, Journal of the Royal Statistical Society, Supplement, 8, 88-90. *Föppl, L. & P. J. Daniell (1913) Zur Kinematik des Bornschen starren Körpers, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 519-529.

Page 52: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

ProductionPeople who have helped.

Birmingham F. J. Daniell Cambridge Jonathan Harrison Jonathan Smith Göttingen Ulrich Krengel Rice Lee Hecht Lisa Moellering Sheffield Helen Mathers Matthew Zawadski MIT Silvia Mejia

Physics Scott Walter Andrew Warwick Control engineering Stuart Bennett

Page 53: “But You have to Remember P. J. Daniell of Sheffield” or Percy Daniell and the British Probability Tradition(s) John Aldrich University of Southampton

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