# Samuel Beatty

### Quick Info

Owen Sound, Ontario, Canada

Toronto, Canada

**Samuel Beatty**was a Canadian mathematician who was the first person to receive a PhD in mathematics from a Canadian university.

### Biography

**Samuel Beatty**entered the University of Toronto in 1903 as an undergraduate. He was to spend the rest of his life studying there and working for the University. After obtaining his undergraduate degree from Toronto, Beatty went on the undertake research for a Ph.D. under Fields's supervision. When Beatty was awarded a doctorate in 1915 he became the first to obtain such a degree from a Canadian university. In fact Beatty was the only student who Fields supervised for a doctorate.

Beatty was appointed as a Lecturer at the University of Toronto after studying for his doctorate. When he was appointed, Alfred Baker was his Head of the Mathematics Department, but in 1918 Baker retired and A T DeLury, who had taught Beatty when he was an undergraduate, became Head. Beatty was promoted to Professor, then in 1934 became Head of the Mathematics Department. In 1936, in addition to his role has Head of the Mathematics Department, he was appointed Dean of the Faculty of Arts and, three years later became a founding member of the Committee of Teaching Staff. He retired from the role of Dean in 1952 and in the following year was elected Chancellor of the University. He held this position until 1959. First let us quote an episode relating to his time as Dean:-

Dean Beatty is remembered for the enormous support he gave to his students, and he earned their deepest appreciation as a result. One of his students, Walter Kohn, who won the 1998 Nobel Prize in Chemistry for his development of the density-functional theory, expressed heartfelt appreciation to the Dean who in 1942 helped Kohn to enrol in the Mathematics Department at the University. Kohn, a young chemist of enormous potential, could not gain access to the chemistry buildings during the war because of his German nationality, and Dean Beatty was instrumental in helping him to continue his studies.Beatty did not publish a large number of research articles. We give a few examples:

*Derivation of the Complementary Theorem from the Riemann-Roch Theorem*(1917), (with Muriel Wales)

*Theory of algebraic functions based on the use of cycles*(1944),

*On the minimum value of the Riemann-Roch expression for order-bases in the large*(1948),

*On the number of conditions to apply to a function*$R(Z, U)$

*to build it on an assigned local order-basis t*(1948), (with N D Lane)

*A symmetric proof of the Riemann-Roch theorem, and a new form of the unit theorem*(1952),

*Upper and lower estimates for the area of a triangle*(1954), and

*Difference methods in the theory of local order bases and their equivalent normalized function bases*(1956). His most famous mathematical idea, however, appeared in a problem he set in the

*American Mathematical Monthly*in 1926. We explain what Beatty did.

Let $R$ be an irrational number greater than 1 and let $S$ be defined by $1/R + 1/ S = 1$. Define two sequences $[nR]$ and $[nS]$ where $n$ runs through the natural numbers and, for any real number $x$, $[x]$ denotes the greatest integer less than or equal to $x$. The two sequences are called 'Beatty sequences' and have the property that every natural number appears in one and only one of the two sequences. For example taking $R$ = 5/π gives the sequences

1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 19, 20, 22, 23, 25, 27, 28, 30, 31, ...

2, 5, 8, 10, 13, 16, 18, 21, 24, 26, 29, 32, 34, 37, 40, 43, 45, 48, 51, 53, ...

R = 6/π gives the sequences
2, 5, 8, 10, 13, 16, 18, 21, 24, 26, 29, 32, 34, 37, 40, 43, 45, 48, 51, 53, ...

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 22, 24, 26, 28, 30, 32, 34, 36, 38, ...

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, ...

while R = 7/π gives the sequences
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, ...

2, 4, 6, 8, 11, 13, 15, 17, 20, 22, 24, 26, 28, 31, 33, 35, 37, 40, 42, 44, ...

1, 3, 5, 7, 9, 10, 12, 14, 16, 18, 19, 21, 23, 25, 27, 29, 30, 32, 34, 36, ...

References to paper investigating Beatty sequences appear in [2].
1, 3, 5, 7, 9, 10, 12, 14, 16, 18, 19, 21, 23, 25, 27, 29, 30, 32, 34, 36, ...

G de B Robinson, the author of [1], was a student at Toronto and was taught by Beatty. He writes:-

Beatty went on sabbatical leave in 1926 - 27 to Wales, attracted by W H Young, so his course on Complex Variables was given by DeLury. Not till 1929 - 30 did I hear Beatty expound the theory of Algebraic Functions based on the Lagrange Interpolation Formula which had been his life work. Perhaps Beatty's greatest contribution to mathematics in Canada was his bringing Brauer and Coxeter to the Department in 1935 - 36.The Canadian Mathematical Society was founded in June 1945 as the

*Canadian Mathematical Congress*. Of all the founding members, Beatty was perhaps the most determined to improve the level of mathematical achievement in Canada through the setting up of such a mathematical society. Beatty was elected as the first President of the Society. The first meeting recorded that the aim of:-

...this congress [will] be the beginning of important mathematical development in Canada.Beatty remained President until 1952, but was honoured with re-elected for a second three year term in 1956-59.

Another venture led by Beatty was organising a series of

*Mathematical Expositions*to be produced by the Department. He wrote the following as a Preface to an early volume which shows his thinking behind the series:-

There are many books dealing in an individual way with elementary aspects of Algebra, Geometry, or Analysis. In recent years various advanced topics have been treated exhaustively, but there is a need in English of books which emphasise fundamental principles while presenting the material in a less elaborate manner. A series of books, published under the auspices of the University of Toronto and bearing the title 'Mathematical Expositions', represents an attempt to meet this need. It will be the first concern of each author to take into account the natural background of his subject and to present it in a readable manner.After he retired as Head of the Mathematics Department and Dean of the Faculty of Arts and Science in 1952 the Samuel Beatty Fund was set up by his friends and former students. Its aim was to:-

... support mathematics education in the province of Ontario through grants, scholarships, and awards.G de B Robinson, ends his tribute [1] with these words:-

... in each role he accomplished great things and won the respect and affection of all who knew him.

### References (show)

- G de B Robinson, Samuel Beatty,
*Canad. Math. Bull.***14**(1971), 489-490. - K B Stolarsky Beatty sequences, continued fractions, and certain shift operators,
*Canadian Math. Bulletin***19**(1976) 473-482.

### Additional Resources (show)

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### Honours (show)

Honours awarded to Samuel Beatty

### Cross-references (show)

Written by J J O'Connor and E F Robertson

Last Update July 2007

Last Update July 2007