Carslaw served the University of Sydney through a period of great expansion. He had precisely that balance of qualities which best fit a man to raise a small department not very different from a Sixth Form to a School of Mathematics with a world-wide reputation for its published work and for the quality of its graduates. He had a vigorous personality of the type around which legends collect; he was a mathematical scholar of distinction with a gift for clear exposition; he was enthusiastic in his mathematics and aroused enthusiasm in his students.
During his tenure of the Chair the University grew from a family affair of a few hundred students to a complex organism of four thousand students in ten Faculties. In particular this period saw the rise of the Faculties of Science and Engineering. In large measure the high standing of Sydney Engineering graduates was due to Carslaw's influence on the shape of the course. His locally famous "little book" bears the title An Introduction to the Infinitesimal Calculus. Notes for the Use of Science and Engineering Students. His first and second year Pass courses had very much the needs of these students in mind. The book is one of those that a lecturer likes to use as a basis for a lecture course; the text is little more than a record of leading results, there are plenty of examples, and the lecturer has plenty of scope for his own elaboration and development of the subject.
Carslaw played a large part in the development of the State (New South Wales) Secondary Education system, in particular in the formulation of the Mathematics syllabus and its examination at the Leaving Certificate. Between 1911 and 1916 he published a series of articles and reports, under such titles as The Teaching of Elementary Mathematics, Secondary Education in Australia, which have done much to give direction and purpose to mathematical teaching in the State right up to the present time.
Carslaw's interests in Mathematics were wide, as a list of his principal works shows:
An Introduction to the Theory of Fourier Series and Integrals and the Mathematical Theory of the Conduction of Heat, first published in 1906 and rewritten and published as two separate volumes in 1921; Plane Trigonometry (1909); Elements of Non-Euclidean Plane Geometry and Trigonometry (1916); Operational Methods in Applied Mathematics (1948, with J C Jaeger). Each of these contains work which is Carslaw's own contribution to the subject. The earlier works were many times reprinted and were standard textbooks for over a quarter of a century. The Fourier Series and the Conduction were translated into Japanese, and the Conduction was reprinted and widely circulated in America during the last War.
The date of publication of the first of the books should be noticed: Hardy's Pure Mathematics was not published until 1908. Carslaw certainly was much influenced by Hardy, who was lecturing in Cambridge while he was there, but Carslaw's book appears to be the first account for undergraduates of the sort of mathematics whose introduction to England will always be associated with Hardy's name.
A bibliography of Professor Carslaw's works will be found in the archives of the Society.
From 1916 onwards he published from time to time articles in the Mathematical Gazette and elsewhere on "Progressive Income Tax", and was consulted by the Commonwealth Treasury on the formulae to be adopted, the aim of the formulae being to give a continuous increase of rate of tax over quite a long range of incomes; such formulae were in use until quite recently.
After retiring from the Chair in 1935, he spent all his time on his small orchard property at Burradoo, eighty miles from Sydney, and only rarely came down to the city. But his interest in Mathematics continued unabated until within the last three years, when failing sight made it impossible for him to continue. After the War, while in his late seventies, he had several years of fruitful collaboration with one. of his students of twenty years earlier, now Professor J C Jaeger. The fruits of this collaboration were the revised third editions of the Conduction and the Operational Methods listed above.
In 1907 Carslaw married Ethel Maude, daughter of Sir William Clark, but she died within the year. Thereafter Carslaw took little part in the social life of the University, but he contributed with his characteristic vigour and vivacity to all other phases of University activity.
He was elected a Fellow of the Society in 1901.
[Much of this notice first appeared in Nature, January 15, 1955, and is reprinted by kind permission of the Editors.]