## The St Andrews Colloquium 1938 and James Gregory Tercentenary

The Edinburgh Mathematical Society held its fourth St Andrews Colloquium in St Andrews from 4 to 15 July 1938. This was a special colloquium at which the 300th anniversary of the birth of James Gregory was celebrated.

## Announcement of the St Andrews Mathematical Colloquium, 1934.

The version we give below of the announcement is from the Application for Membership Leaflet:

Under the auspices of the Edinburgh Mathematical Society, a Mathematical Colloquium is being held in the University Hall, St Andrews, Scotland, from July 4th to 15th, 1938.The year 1938 is the three hundredth anniversary of the birth of James Gregory, who held in succession the Chairs of Mathematics in St Andrews and Edinburgh Universities; and the occasion will be celebrated in the following ways in connection with the Colloquium.

- The Royal Society of Edinburgh will hold a commemoration of James Gregory in their Rooms at 22 George Street, Edinburgh, on the afternoon of Monday, July 4th. Members of the Colloquium are specially invited to attend this meeting, which will be regarded as the opening meeting of the Colloquium, on their way to St Andrews.

- The University of St Andrews will hold a Graduation at which Honorary Degrees will be presented to a number of eminent mathematicians. This will take place on Tuesday, July 5th, and will be followed by an evening reception.

- Professor H. W. Turnbull, F.R.S., Regius Professor of Mathematics in the University of St Andrews, will lecture during the Colloquium on
Recent Discoveries in the History of XVII Century Mathematics, making special reference to the life and work of Gregory.

The following Courses of Lectures have been arranged:

Invariant Matrices and the Symmetric Group.

By A. C. Aitken, F.R.S.

Abstract groups of finite order: representations of groups: groups of permutations: groups of linear transformations: reducibility: group characters: invariant matrices: applications.

Analytic Deformations and Auto-equivalent Functions.

By G. D. Birkhoff.

Professor of Mathematics in Harvard University, Cambridge, Massachusetts.

Theory of analytic deformations: theory of the corresponding linear functional equations: auto-equivalent and invariant functions: groups of deformations and their invariant or auto-equivalent functions.

The Interactions between the Elementary Particles of the Universe.

By E. T. Whittaker, F.R.S.

The classical concept of force between particles: its replacement in relativity and quantum mechanics by other descriptions of interaction: Einstein's theory of gravitation: quantum theory of the interaction of electrified particles: exchange interaction: the neutrino: recent work on the interactions of neutrons and protons: the theories of Eddington and E. A. Milne.

Aspects of Mathematical Biology.

- By W. O. Kermack, D.Sc., LL.D.

Mathematics of Population Growth.

The Malthusian parameter: concept of present values: integral equations of the Volterra and Fredholm types. Mathematical Epidemiology. Deduction of the integral equations: the steady state: threshold values: stability problems. Statistics and Biology. The problem of sampling: method of inverse probability: method of fiduciary probability: other methods: the general nature of scientific knowledge.

- By I. M. H. Etherington, B.A., Ph.D.

Volterra's Theory of Competing Species.

The fundamental equations and their integration: fluctuations of competing species: principles of conservation of "energy" and least "action."

In addition to the above Courses, single lectures will be arranged; also informal talks, and discussions on the lecture courses.A paper on

Neglect of Form and Law in School Algebrawill be offered for debate by George Lawson, M.A., President of the Edinburgh Mathematical Society; and other discussions of School Mathematics will be arranged if there is demand for them.Arrangements will be made for golf, tennis, excursions, musical evenings, etc.

By the courtesy of the St Andrews University Court, the Colloquium will be held in the University Hall. Members of the Colloquium (and their relatives or friends) may stay at the Hall, which has been reserved entirely for this purpose.

The fee for the Colloquium is £1 5s. (or 12s. 6d. for students), of which 5s. is payable as the registration fee when application is made. The cost of board and lodging for the period of the Colloquium (dinner on July 4th to breakfast on 15th) will be £5 15s. per head (£3 15s. for children, under 14, of members); some reduction will be made for shorter periods.

Application should be made in the annexed form as early as possible, and in no case later than 15th June. Early application is particularly advisable in the case of those who propose to stay at the Hall, as the accommodation is limited.

St Andrews, with its bracing sea air and bathing, its famous golf courses, and its many interesting mediaeval buildings, is a favourite holiday resort.

## The Press Release.

The following press release announced the coming event on 1 April 1938:

ST ANDREWS MATHEMATICAL COLLOQUIUM.

James Gregory Tercentenary.The 300th anniversary of the birth of James Gregory, who held in succession the Chairs of Mathematics in St Andrews and Edinburgh Universities, will be celebrated at a Mathematical Colloquium to be held in St Andrews from July 4 to 15, 1938, under the auspices of the Edinburgh Mathematical Society. By courtesy of the St Andrews University Court, the Colloquium will be held in the University Hall, where members of the Colloquium will stay with their relations and friends. Short courses of lectures on subjects of pure mathematics, mathematical physics, and mathematical biology will be given by Dr A C Aitken. F.R.S.; Professor G D Birkhoff, of Harvard University, Massachusetts; Dr W O Kermack, LLJD., and Professor E T Whittaker, F.R.S. Lectures on the work of Gregory and his contemporaries will be given by Professor H W Turnbull, F.R.S. A discussion of school mathematics will be opened by Mr George Lawson, president of the Edinburgh Mathematical Society. The opening meeting, which will be a commemoration of James Gregory, will be held in conjunction with the Royal Society of Edinburgh in their rooms at 22 George Street, Edinburgh, on the afternoon of July 4. Members of the Colloquium will also be able to attend a graduation in the University of St Andrews, at which honorary degrees will be presented to a number of eminent mathematicians. This will take place on July 5, and will, be followed by an evening reception. The office of the hon. secretary of the Edinburgh Mathematical Society is at 16 Chambers Street.

## Another Press Release.

The following press release described the courses at the colloquium:

GREGORY TERCENTENARY.

Colloquium at St Andrews.

ADVANCE OF MATHEMATICS.In connection with the James Gregory tercentenary the Edinburgh Mathematical Society are holding a Colloquium in St Andrews this week and next. Professor H W Turnbull is delivering a series of lectures on the work of James Gregory. During the last seven years of his life, said the speaker, he refused to publish any of his work, except one trifling little thing which he put in at the end of a book, entitled,

The New Art of Weighing Vanity. The reason was that he was waiting for Newton to publish his works.In his lectures Professor Turnbull has been laying bare the proof of what Gregory actually discovered. For example he had evolved a theory of the calculus that we taught in schools today, except for the notation.

The interest of his work lay in the fact that he had thought of those things before anyone else except Fermat in France and Dr Pell in Switzerland and in England. The next advance in that theory did not take place until a century later at the time of Lagrange.

PROFESSOR WHITTAKER'S COURSE.Professor E T Whittaker Is giving a course on "The Interactions between the Elementary Particles of the Universe." The first lecture dealt with the modifications which the Newtonian theory has undergone in consequence of the discovery of Relativity. The second lecture, yesterday morning was concerned with the "exchange interaction" which accounts for the binding of two atoms of hydrogen into a hydrogen molecule.

Professor M Fréchet, of the university of Paris, who recently published a book on "The Definition of Probability" in two lectures expounded the diverse definitions which have been given of the probability of an event and has compared their respective values. Dr A C Aitken, of Edinburgh, dealt with "Invariant Matrices and the Symmetric Group." Dr W O Kermack. Edinburgh, spoke on "Aspects of Mathematical Biology."

## Report of George Lawson's Lecture.

One of the papers read to the Colloquium (on Saturday 9 July) was by George Lawson, president of the Edinburgh Mathematical Society. The following report of the lecture appeared in The Scotsmanon Monday 11 July:

Teaching of Algebra in SchoolsAt the St Andrews Mathematical Colloquium, which is being held in connection with the James Gregory Tercentenary at University Hall, St Andrews, Mr George Lawson, president of the Edinburgh Mathematical Society, on Saturday night submitted for debate a paper on "Neglect of Form and Law in School Algebra." He said that by the British Association papers last year, it was indicated that algebra had, during the last 30 years, developed on the lines of Sir Percy Nunn's classical exposition. That same development was indicated also in 1935 by the Mathematical Association's special pamphlet on the teaching of algebra in schools. In criticising that development, Mr Lawson referred to another development which seemed possible 30 years ago, and said it was based largely on a book quoted by Sir Percy Nunn himself - namely, that of Barnard and Child. Mr Lawson took up the position that algebra at that time took the wrong road, and he advocated a modification of the Barnard and Child position.

## Report on the St Andrews Mathematical Colloquium, 1938.

A report on the Colloquium, written by I M H Etherington, appeared in The Mathematical Gazettelater in 1938. The full reference is I M H Etherington, Edinburgh Mathematical Society: St Andrews Colloquium,The Mathematical Gazette22(252) (December 1938), 482-484. We present below a version of this report:

EDINBURGH MATHEMATICAL SOCIETY:

ST ANDREWS COLLOQUIUM.This Colloquium, held at St Andrews from July 4 to 15, was in every way as successful as its quadrennial predecessors. A hundred and ten persons attended, including about twenty wives and daughters of mathematicians; including also twenty-one professors of mathematics from the universities of Great Britain, Eire, France, Holland, Denmark, and the United States.

Halfway through, a professor was heard to boast that he had talked no mathematics so far, on the other hand, many members of the Colloquium found it a grand opportunity for talking fruitful shop. The arrangement of the timetable encouraged the morning coffee habit; so the café gardens of the town, and in lesser concentration the roads and walks about the place, were the scene of much deep talk on the foundations of probability, theories of the universe, and the personalities of mathematicians.

The Colloquium associated itself in various ways with the name of James Gregory, born 300 years ago, who held in succession the Chairs of Mathematics in the Universities of St Andrews and Edinburgh. The opening meeting on July 4 was held in Edinburgh in conjunction with the James Gregory Tercentenary Meeting of the Royal Society of Edinburgh. Two events of this meeting may be recorded here: the presentation to the R.S.E. of the portrait of its President, Sir D'Arcy W Thompson, painted by Mr David S Ewart, A.R.S.A. and the presentation to Professor H S Ruse of the Society's "Keith Prize" for his work on the geometry of Dirac's equations.

Next day the University of St Andrews marked Gregory's tercentenary by conferring the Honorary Degree of LL.D. on five distinguished mathematicians, members of the Colloquium being invited to the celebrations. The graduands were Professor G D Birkhoff (Harvard), Professor A W Conway (Dublin), Professor Otto Neugebauer (Copenhagen), Professor R Weitzenböck (Amsterdam), and

in absentiaProfessor V Volterra (Rome). The ceremony took place in the University Library, where Gregory used to work; here also some MSS., instruments and other relies of Gregory were exhibited.Professor H W Turnbull, Gregory's present successor as Regius Professor of Mathematics at St Andrews, lectured on these two occasions, and again in the course of the Colloquium, on Gregory's life and work, and especially on his many unpublished discoveries. Professor Turnbull, has made an exhaustive study of Gregory's MSS. and proved that he should rank with Barrow, Newton and Leibniz as a founder of the Calculus.

Of the four main courses of lectures, two were on subjects of pure mathematics. Professor G D Birkhoff lectured on

Analytic Deformations and Auto-equivalent Functions, and Dr A C Aitken onInvariant Matrices and the Symmetric Group. Ananalytic deformationof a functionf(x) in the neighbourhood ofx= infinity means replacing the independent variable by an analytic function of itself,x' =f(x), which has a simple pole at infinity. All such deformations form a group. The functionx^{k}has the property of being multiplied by a function which is analytic and non-zero at infinity when any deformation of the group is applied. Other functions, such asG(x), have this property only for a smaller group of deformations; in this case

Such functions arex' =x+ any integer.

auto-equivalent,and Professor Birkhoff classifies them into eight types according to the defining group. The theory, which promises to unify much of modern analysis in one comprehensive sweep, is to be published by the Institut Henri Poincaré. Unification was also an achievement of Dr Aitken's course on algebra, which led through symmetric functions, finite groups, determinants and permanents, Young's standard tableaux, and Schur's invariant matrices to the foreshadowing of a master theorem which would combine them all.On the side of applied mathematics, Professor E T Whittaker gave five lectures on "The Interactions between the Elementary Particles of the Universe". The modern picture of these particles differs immensely from the classical views of Newton and Maxwell. Professor Whittaker covered this immensity, and described the stages in which the transformation took place, through relativity, the early quantum theory and its developments up to the latest speculations on heavy electrons and cosmic rays.

The fourth main course was entitled

Aspects of Mathematical Biology. Dr W 0 Kermack discussed the integral equations which arise in the mathematics of population growth and of epidemiology. Dr I M H Etherington surveyed Volterra's mathematical theory of the struggle for life and its analogies in classical dynamics. It was impossible not to admire Dr Kermack's masterful exposition of his complicated subject matter, aided by lantern slides displaying his formulae but without the gift of sight.Professor M Fréchet lectured and introduced a discussion on the various definitions of probability. He sketched the views of Laplace, de Mises, Wald and others, and described in more detail the "modernised classical definition" of Neyman and Kolmogorov. The discussion was noteworthy for Professor Whittaker's vigorous defence of the classical (Laplace's) point of view against all comers.

Mr George Lawson, the President of the Edinburgh Mathematical Society and Chairman of the Colloquium, addressed himself mainly to school teachers, who formed a not inconsiderable proportion of the membership. Mr Lawson regretted the lack of sincerity in much modern teaching, and urged the importance in teaching algebra of noticing its formal aspect algebra is the study of the

formsin which numbers cooperate Professor Otto Neugebauer entertained us for half an hour on the subject of Babylonian astronomy, and on the differences between Babylonian and Egyptian science.Finally, I must not forget to mention Professor Birkhoff's fourth lecture, which was not on auto-equivalent functions but on

The Mathematical Theory of Art. He claims to have found a formula for the aesthetic value of any work of art in its formal aspects, and to have devised rules by which it can be applied with demonstrable success in certain cases of special simplicity. The theory, which is little known on this side of the Atlantic, was received by a crowded audience with much interest and a certain scepticism.Our hosts, Professor and Mrs Turnbull, entertained us and many visitors at a reception one musical evening; and many less formal

moments musicauxoccurred. An incomparable five minutes from Sir D'Arcy Thompson, illustrated with geometrical models, happened unheralded but much applauded in the middle of one informal concert. A bald mention of golf, tennis, rounders, dancing, chess, and an excursion to the Highlands must conclude this account.I M H ETHERINGTON

JOC/EFR April 2016

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