As soon as Jackson was of walking age, he became his father's companion on long hikes and his father's interest in science made these excursions essentially field trips for the study of botany, zoology, and geology. Thus, very early, Jackson acquired from his father a vast store of miscellaneous information and habits of observation that he never outgrew. At an early stage, he started reading the science textbooks in his father's library.

Guided partly by natural inclinations, perhaps, and partly by recollection of a course on methods of approximation which I had taken with Professor Bôcher a few years earlier, I committed myself to one of the topics which Edmund Landau had proposed for a thesis, an investigation of the degree of approximation with which a given continuous function can be represented by a polynomial of given degree. When I reported my choice he said meditatively "... Das ist ein schönes Thema, ich beneide Sie um das Thema ... Nein, ich beneide Sie nicht, aber es ist ein wunderschönes Thema." Ⓣ

Jackson's thesis answered a prize question which had been proposed earlier by the Göttingen faculty: "Would it be possible to improve on results (on approximation by polynomials) previously obtained by de la Vallee-Poussin and Lebesgue?" In a complimentary statement awarding Jackson the prize, we find the remark: "The author ... has enriched the science with valuable results, all in competition with mathematicians of the first rank."

It is no reflection on Jackson's later achievements to remark that his thesis was one of his most important pieces of mathematical research. The fact that his results improved on those of very famous mathematicians gives sufficient assurance of the quality of his thesis, and Landau was amply justified when he labelled its topic "ein wunderschönes Thema." Jackson thus had the rare good fortune to produce, in his first major effort, results which were fundamental for the development of a major field.

As an outstanding characteristic, he enjoyed his associations with people. In particular, his congenial wife and family were a continuing source of pleasure and inspiration to him.

In 1925, Professor Jackson lectured at the American Mathematical Society's Colloquium on certain aspects of the theory of approximation, i.e. on the theory of approximating to a function of a real variable by means of a finite sum of functions of given form. This theory covers a vast field, and, in his lectures, Professor Jackson chose certain topics in a few corners of this field in which he is personally interested and where he has made valuable contributions.

The title of this volume is an abbreviation for the more properly descriptive one: 'Topics in the Theory of Approximation'. It is an account of certain aspects and ramifications of a problem to which I was introduced at an early stage, and which has given direction to my reading and study ever since.

The reader will enjoy it as interesting, stimulating, extremely well written, and beautifully printed. Many a worker will be inspired by the pages devoted to Fourier and Legendre series and to Chebyshev-Hermite polynomials to try his hand in obtaining similar results for other classes of orthogonal functions, and in particular, for other classes of orthogonal Chebyshev polynomials.

Under the circumstances, 'rigour' in the sense of literal completeness of statement has been out of the question. It is hoped, however, that the reader who is familiar with the methods of rigorous analysis will be able without any difficulty to read between the lines the requisite supplementary specifications, and will find that what has actually been said is entirely accurate in the light of such interpretation.

The task which the author has set himself is no easy one, but he has succeeded in producing a sound and very readable book well suited to the purpose he had in view.

... measures up fully to the expectation of the reader who is acquainted (and who is not?) with the lucid and elegant presentation - oral and written - of Dunham Jackson.

The exposition is well organized; much of the formal work is beautifully handled; the proofs are extremely clear. It will not be surprising if this turns out to be one of the most popular of the Carus Monographs. The Association is indeed fortunate to be able to add to the Carus series an exposition by one of the world's foremost authorities on orthogonal polynomials.

A proper balance between formal applications and rigorous analysis has been maintained, and the approach in general has the clarity and elegance that one has come to expect from this particular author.

Jackson was a man of high ideals, extremely unselfish, and very conscious of his responsibilities, not only in strictly personal affairs but also with respect to his profession as an educator and his duties as a citizen. He was intensely devoted to the field of mathematics, had thought deeply of its place in advancing human welfare, and was fluent in expressing his convictions on such matters.

What is the purpose of a higher course in mathematics? To teach facts or methods? Should a course be simply a succession of theorems with proofs? Or, is it undesirable to try to prove everything? Does a mathematician read and make absolutely clear to himself all the work on which his own original work is based, so that he can assert, of his own knowledge, that the preceding work is true? Or, does he leave something to the accuracy of his predecessor? This is a point on which information ought to be gained by inquiry. Is the purpose of proofs in a higher course to teach methods which will be useful to the pupil or to assure him that every point is based on a rigorous foundation? If the answer to these questions is different for different courses or for different parts of a course, I think it would be well for the teacher to have in mind just what point of view he is adopting at each stage.