Vaidyanathaswamy studied in Edinburgh with E T Whittaker, then visited Cambridge and worked with the geometer, H F Baker.

I certify that Mr R Vaidyanathaswamy has spent six terms at Research Work under my supervision in the University of St Andrews, and that he has fulfilled the conditions of Ordinance No. 16 (St. Andrews) and that he is qualified to submit the accompanying Thesis in application for the Degree of Ph.D.

I matriculated in the University of Madras in 1910 and followed a course leading to graduation in Arts; subsequent to it I spent four years as a Research student in the same University.

In November 1923 I commenced the research in Algebraic Geometry which is now being submitted as a Ph.D. Thesis.

I am the holder of a Scholarship from the University of Madras and one from the Madras Government; Also of a grant from the British Department of Scientific Research.

I was in attendance as a research student at the University from November 1922 to July 1924.

I qualified for the degree of M.A. with third class honours in mathematics, in the University of Madras, in 1917; this degree has been accepted by the Senate as equivalent to the degree of M.A. or B.Sc. with honours in the university of St Andrews.

The theory of Apolarity which is a principal source of inspiration in Algebraic Geometry, was in great vogue about the middle of last century with a school of German Mathematicians (Clebsch, Meyer, Waelsch, Reye &c.) as a natural consequence of the great advance of Invariant-theory at that time. Some of the main lines of the subject were then laid down, and its more obvious features worked out; but it appears to have fallen since into comparative neglect and disuse, cropping up only on occasions.

Binary Apolarity which is the special part of Apolarity with which we are concerned here is a subject which is much more limited in scope, though it makes up for its lack of extent by its corresponding elegance and prettiness. Its great geometrical application is to Rational Curves in general, and in particular to the Norm Curve, which being the prototype of all rational curves may be taken as the rational curve par excellence. Interpreted by means of the Norm Curve, the relations arising out of Apolarity translate themselves into a Binary Geometry of the space of n dimensions. The linear subspaces correspond to linear systems of Binary forms, and speaking generally there is a parallel march between Binary Algebra on one side and the Projective Geometry of the Norm curve on the other.

Some of the ideas in this work were suggested by Henry F Baker, University of Cambridge.

From this date till his retirement in 1952, he served Madras University conducting and guiding research and lecturing on many basic modern disciplines like Logic, Set Theory, Lattice Theory, Topology, etc.

Since geometry was the subject which received the greatest attention in South India at that time, it is not surprising that many of his early papers and a good deal of his later work should have a geometrical basis. In fact, references to the pedal line property of the triangle and its generalisation to a cyclic n-gon, normals from a point to a conic and to a quadric, porisms of triangles and polygons inscribed in one conic and circumscribed to another, the rational normal curve and incidence results in n-dimensions and similar topics figure largely in his papers. But he was not interested in studying generalisations suggested by such problems either by the plodding methods of analytical geometry or the blind gropings of classical pure geometry. At quite an early stage he became intensely interested in the theory of forms binary, and multiple binary forms and their invariant theory, and made an excellent study of classics like 'Algebra of Invariants' by John Hilton Grace and Alfred Young and 'Invariententheorie' by Roland Weitzenböck, and acquired a good mastery of the methods of symbolic calculus. Consequently, these problems from elementary geometry were raised to a higher dimension by being transformed into problems on correspondences, transformations and invariant theory.

His general method of procedure appears to be something like the following. First, study all the available literature on a topic, then reformulate a geometrical problem from the point of view of correspondence theory and allow the work to proceed on natural lines as far as it will go. The underlying geometrical basis allows these form-theoretic results to be interpreted in terms of geometry and suggests further work in geometry. This work in geometry, in turn, suggests further investigations on the form-theoretic side. This interplay is pushed as far as possible and all the suggested lines of work carried out before the results are embodied in a paper of great elegance. His papers could be studied ab-initio and embodied all the known results and several new ones in a unified manner.

The distinguishing features of Dr Vaidyanathaswamy's treatment in this heavily worked field are the effective use of canonical forms, the parametric treatment of Poncelet's theorem and the explicit use of the operational calculus. The study of these various special cases led him to formulate concepts for the general (m, n) correspondence.

The concept of unitary divisors was introduced by the Indian mathematician Ramaswamy S Vaidyanathaswamy (1894-1960) in 1931. He called them "block factors". The term "unitary divisor" was coined by Eckford Cohen (1960).

This book which has been in the press during the last two years, has grown from the courses of lectures delivered for several years in succession in the Department of Mathematics of the university of Madras; as the first part of the whole treatise, it is devoted to the basic topological ideas. It is hoped that it will be possible to bring out the second part, which is under preparation, in the course of the next year.

In writing the book I have not hesitated to make use wherever necessary of the classical treatises on the subject - Hausdorff's 'Mengenlehre', Kuratowski's 'Topologie', and Alexandrov-Hopf 'Topologie'. A special feature of this work is the large number of exercises which have been included in every chapter; these are intended generally to furnish applications and illustrations of the methods of the text, and occasionally as problems for the ambitious reader. Many of the results in these exercises have been taken from the above classical treatises, many have been adapted from original papers in the mathematical journals, and the rest are original; as a rule, in the case of striking results, I have indicated the author and the source, except where they are well-known, or original.

If a new approach to the subject can be claimed for this book, it should be described as one which seeks to exploit partial order, wherever it occurs in topological situations.

This book is the first part of a two volume work based on lectures given by the author at the University of Madras. A feature of the treatment given is its extensive use of partial ordering. A chapter is devoted to the theory of partially ordered sets and lattices. In it various notions are introduced which are used later on for giving lattice theoretic definitions and interpretations of various topological notions. For example, a bicompact space turns out to be a topological space whose lattice of open sets "has a jump at one" and the interiors of closed sets appear as the "normal" elements of this lattice.

There are eleven chapters. The first two contain preliminary material on the algebra of sets and related topics and a third the discussion of partial ordering referred to above. The fourth chapter takes the Kuratowski closure axioms as a point of departure and introduces such notions as open set, closed set, interior, boundary, etc. It concludes with a section each on relative topology in subsets and on metric spaces. Chapter V begins with Hausdorff's neighbourhood postulates and after some indications concerning their relationship to the Kuratowski axioms proceeds to discussions of strength of topologies and the various countability and separation axioms. Chapter VI deals with various notions definable in an especially natural way in terms of properties of the lattice of open (or closed) subsets of a topological space and in particular with compactness and bicompactness. In chapter VII functions from one topological space to another are considered. The principal topics treated include continuity, resolution spaces, complete regularity and the extension of continuous functions. Chapter VIII consists primarily of a detailed discussion of the theory of the "localization of hereditary additive properties" of Kuratowski and the related generalisation of the notion of derived set. The properties of being finite, of being non-dense and of being of the first category are given particular attention. Chapter IX is concerned with finite and infinite direct products of topological spaces. Topologies for the products different from those usually studied are also given attention. A section of the chapter is devoted to connectedness but oddly enough the theorem that the direct product of connected spaces is connected is given only in a very special case. Chapter X deals with the theory of metric spaces with special emphasis on the non-topological notions of uniform continuity, completeness and total boundedness. Chapter XI presents an exposition of a variant, with emphasis on lattice theoretic concepts, of Fréchet's theory of "convergence spaces."

The book contains a very large number of exercises which anyone teaching a course in the subject should find quite useful.

... played a leading role in the advancement of mathematics in India for over three decades. He created the consciousness of a cohesive national mathematical community long before India's Independence, through all the years of civil strife and even of war. His influence endures in the work of his many students, and their students in succession ... His achievements are not confined to a few outstanding papers but spread throughout his career, of rigour and development, in the service of mathematics. He modernised the outlook, and widened the perspective, of his colleagues and students as never before. His name commanded respect in all corners of India not just because of his high professional competence and irreproachable personal integrity; he was looked up to as a man of culture endowed with spiritual merit. It is with gladness and gratitude that those of us who had the good fortune to work with him as our leader whisper his name to ourselves on this occasion, as though touched by his presence.

As a man, he was simple, even austere in his dress and habits. He was a strict vegetarian, never drank, rarely smoked and indulged in few luxuries unless betel and tobacco chewing and novel-reading be included among luxuries. Besides mathematics, he was keenly interested in Karnatic music and in yogic sadhana. He was a keen student of Sri Aurobindo's Integral Yoga and was for some time Editor of a journal, Advent, devoted to the exposition of Aurobindo's philosophy. He studied the Vedas in their original Samskrit text and believed with Aurobindo that there were deep inner meanings associated with them which modern Indians should seek to unravel. In fact, right up to a brief period before his death, he was giving lectures every week on the interpretation of some of the Vedic texts. Warm and loveable, dignified and cultured, there was nothing narrow in his outlook. He was ever ready to discuss difficult points and give helpful guidance to students whether his own or working elsewhere. He was not only a great mathematician but, what is rarer, a great man.