- An Early Method for Summation of Series.
The Mathematical Gazette 15 (205) (1930), 5-11.
The method given by Euler for the transformation of series was used by him to obtain "sums" for various divergent series, and recently Konrad Knopp [Math. Zeit. 15 (1922), 226; Math. Zeit. 18 (1923), 125] has written extensively on the "Euler summation process" derived from this transformation. But it does not seem to have been remarked that an interesting variation of this method is given by Charles Hutton in the first volume of his Tracts, published in 1812. In Tract 8, "The Valuation of Infinite Series," written in 1780, Hutton gives an account of his method, claiming to have arrived at it independently of Euler's work, and employing it to approximate to the sums of series whose terms alternate in sign. When the proposed series converges, we have an easy and rapid way of determining its sum, and in the case of a divergent series, a number is determined which would nowadays be recognized as a conventional sum. Very little attention seems to have been paid to Hutton's paper and his name is apparently not mentioned by modern writers on this subject. Augustus De Morgan [Diff. and Int. Calculus (1842), chap. 18, "On Interpolation and Summation"] refers to the "remarkable method" of Hutton, but he applies it only to convergent series, and makes no reference to its application to oscillating series. This is noteworthy in view of the remarks made by De Morgan at the end of his Chapter 19 on "The Transformation of Divergent Developments," but it might be conjectured that even so acute a logician as De Morgan had not clearly perceived, at any rate at the time of writing his "Calculus," that the word "sum" applied to any infinite series is being used in a conventional sense. It is not to be expected that Hutton, writing in 1780, had any precise notions on convergence. In Tract 7 on "The Nature and Value of Infinite Series," he defines a convergent series as one in which the absolute magnitudes of the terms form a decreasing sequence and a divergent series as one in which they form an increasing sequence; a series of the type
a - a + a - ...is said to be neither convergent nor divergent but neutral. The theory given by Hutton turns on a fallacy connected with the use of this series. For this reason, although the theory is interesting and suggestive, possibly the most valuable part of his work is the neat and systematic setting-out of the computations.
- The unification of algebra in schools. An Introductory Survey of the Present Situation.
The Mathematical Gazette 21 (246) (1937), 314-317.
The teaching of algebra has never been dominated by a single work in the way in which Euclid for so long a time dominated the teaching of geometry. For that reason, perhaps, the revolts in geometry have corresponded to quiet reforms in algebra, and the problems involved, though equally important, have never received the same amount of attention. On the teaching of the subject in schools, excellent books exist; there is Sir Percy Nunn's classic, and more recently, a book by Clement Durell, and much relevant matter in Charles Godfrey and Arthur Warry Siddons, 'Teaching of Elementary Mathematics' (1931). Even so, it is not easy to obtain a comprehensive view of the subject in its development through the school to the university, nor has the impact of recent trends in research work on the demands of the universities been fully investigated. As part of this morning's discussion will be concerned with this impact, it might perhaps be worth while clearing up one point. The idea that the university teacher spends the whole of his time in training professional mathematicians is an illusion; in the modern universities he spends a good part of his time training those whose interest in mathematics is subsidiary to their interest in other subjects or to their interest in the process of obtaining a degree. When we speak of what the universities expect from the schools, we mean simply what the universities rightly expect from those whose last years at school have been mainly given to mathematics. Most university teachers understand that this question, though naturally the one which most interests them, is by no means the whole or even the larger part of the problem of the teaching of mathematics in schools.
- The Mathematical Gazette: Our History and Aims.
The Mathematical Gazette 30 (291) (1946), 186-194.
The Association owes its existence to one man, though he died some 2000 odd years before its birth. Euclid wrote (among other things) a masterpiece, 'The Elements of Geometry', and he wrote it for the cream of the adults of a race marked by a peculiar, perhaps a unique characteristic: the Greeks liked thinking. They delighted in abstract logical thought for its own sake, and admired masters of geometry as we admire a Hobbs or a Verity. 'The Elements', with its orderly system, its brilliant logical triumphs in the theory of parallels and the theory of proportion (for the more Euclid's treatment is examined, the more impressive it becomes, in spite of gaps which later work has revealed), was a peak of Greek culture. How quite it came to be inferred from this that the book was one pre-eminently suited for the education of small boys in Victorian England, how it came to pass that the small boy learned his Euclid as he learned his list of the Kings of Israel and Judah, how it became, accepted as incontrovertible that immense mental and spiritual benefits must result from these exercises, forms a curious chapter in the history of education in this country. Even to-day one can find teachers who believe that Euclid is the best thing in the world for the training of small children. It would perhaps be fairer to compare Euclid with Rabelais - both are masters, but neither is really suitable for young children. By the middle of the nineteenth century, some few daring heretics among our teachers were beginning to see this, and even beginning to say it; and these heretics, meeting at Rugby in 1870, founded an 'Association for the Reform of Geometrical Teaching', and thereby struck the first blow in the revolution in the teaching of school mathematics, a revolution still going on. Of course, the reformers made mistakes; they were bigoted, they were inclined to refuse any virtue to Euclid, their substitutes for him were far from perfect. They worked under great difficulties; they had opposed to them the vested interests and the enormous inertia of the existing system ...
- Printer's Ink and the Teacher. Presidential Address to the Mathematical Association, 4th January, 1954.
The Mathematical Gazette 38 (324) (1954), 81-89.
Let me begin then with a story concerning that great American mathematician and physicist, Josiah Willard Gibbs. Gibbs, it is said, was a most reticent man, and during the long period of his service on the Senate of the University of Yale, he spoke only once. The occasion was a demand from the department of modern languages for more facilities, these to be provided by a sacrifice on the part of the department of mathematics, the demand concluding with an eloquent peroration depreciating the value of mathematics in comparison with the educational and cultural status of languages. All eyes turned to the taciturn Gibbs. Eventually he rose, pronounced one brief sentence and sat down again. What he said was: "Mathematics is a language." To anyone concerned with the process of embodying the abstract ideas of mathematics in the cold permanence of type, this view of the subject must become fundamental. If I were asked the question "What is mathematics?" I could find no better answer than these words of Gibbs, though perhaps, lacking his superb control of his tongue, I might be tempted to qualify and amplify this definition. The constant, almost daily work of trying to see how best to convey mathematical ideas from one mind, through the medium of print, to another, leads to a definition of mathematics, not indeed entirely comprehensive but near to the heart of the matter. Mathematics is the language of abstract rational thought. If reproach is levelled against such high-brow adjectives, it may be remarked that even the most fanatical devotee of "practical" mathematics can hardly deny that in determining, say, the area of a rectangle, he has to form the product of two numbers, an abstract process employing abstract ideas, and that, if he should be anxious to determine the true area, he will do well to adhere to rational methods of multiplication. Mathematics is the natural mode of expression for abstract logical thinking.
- Professor Eric Harold Neville, M.A., B.Sc. On the teaching committee.
The Mathematical Gazette 48 (364) (1964), 136-139.
The Mathematical Association was founded in 1871, as an 'Association for the Improvement of Geometrical Teaching', by teachers who had lost faith in the educative value of learning by rote the words of Euclid's sacred text; indeed, one suggested name was the Anti-Euclid Society. Progress was at first slow, until in the early years of this century Oxford and Cambridge agreed to accept, in their entrance requirements, any system of geometry which exhibited some logical coherence. This encouragement evoked a number of new school geometries, some good, some not so good, but generally differing from Euclid only by omissions and some small rearrangements. The more ardent reformers were far from satisfied, and demanded a much more radical revision of the geometry syllabus, to encourage geometrical intuition and the recognition of key theorems, while laying little stress on the formation of a chain of logical steps until the need for this should be evident to the pupil himself. Of these radicals, T P (later Sir Percy) Nunn was one, and it was natural that he should seek to enlist the most brilliant of his former pupils under the reformers' banner. The opportunity came in 1922, when the General Teaching Committee of the Association appointed a strong sub-committee charged to draw up a report on the teaching of geometry in schools. Neville had recently been elected to the chair of mathematics at University College, Reading, (now the University of Reading) and problems of teaching method had begun to engage his attention. He accepted Nunn's call with enthusiasm and threw himself wholeheartedly into the work as chairman of the committee, coming to the task not only as an alert and inspiring teacher but also as a professional mathematician of acknowledged skill and logical acumen.
- George Boole (1815-1864).
The Mathematical Gazette 48 (366) (1964), 373-378.
An American mathematician, celebrated for a love of epigram tempered by some regard for truth, has asserted that George Boole has the sour distinction of being the most severely under-rated mathematician of the 19th century. There is some substance in this judgement. During Boole's lifetime, his work was recognised and admired in this country, but British mathematics was then very insular, paying small attention to the great progress being made in Europe, and, reciprocally, being little regarded by Continental mathematicians. Boole's reputation was localised, and, moreover, it rested very much on what we should now consider to be the less valuable of his contributions to mathematics. The title of his greatest work, The Laws of Thought, does not suggest a strictly professional study of pure mathematics but rather a digression into philosophy and metaphysics. Yet Bertrand Russell has declared that Boole, in this work, was the first person to discern clearly the essential characteristics of pure mathematics, and today, when Boolean algebra is a topic for school texts as well as for research journals, we can see that Boole's work, under-valued at the time of its appearance, was an essential step in the development of pure mathematics as an abstract and axiomatic structure.
- Institute; Joint Council; Association. A Talk Delivered at the Annual General Meeting of the Association in April, 1965, at Oxford.
The Mathematical Gazette 49 (369) (1965), 262-265.
We really should discuss what the teacher of mathematics can do and ought to do for the Association, for the Institute and for the Joint Council. Professor Semple will be mainly concerned with the Institute and the Joint Council, and I will confine myself mainly to the Association; but perhaps the various domains should be clarified. The line of distinction between the Association and the London Mathematical Society, for instance, is clear: the London Mathematical Society is primarily interested in mathematical research, in the discovery and publication of new mathematics, particularly of recent years in pure rather than applied mathematics, and it is our premier research society. Here the Association does not compete. It is true, and is worth remembering, that some very distinguished research workers have first appeared in print in the pages of The Mathematical Gazette, for instance the late Prof G N Watson; I hope we shall have more instances of this from time to time, but it is not our primary purpose. The Association is concerned with the teaching of mathematics; it began as a revolt against outmoded and restrictive syllabuses and methods in geometry; it extended its field to all the mathematics of the upper school, and thence to the teaching of mathematics at all levels. Our field is not that of mathematics alone, nor that of teaching alone; it is the combination, the teaching of mathematics. We do not want a good way of teaching bad mathematics, or a bad way of teaching good mathematics; we need to teach good mathematics in a good way, and nothing less will serve. The Association of Teachers of Mathematics is perhaps more concerned with teaching, our Association more with mathematics; but there is plenty of room for both societies. The Institute is of course concerned with the teaching of mathematics, but this is not its primary purpose. It seems that the Institute will be chiefly engaged in furthering the uses of mathematics. For this purpose, it must of course keep in touch with the London Mathematical Society, to know what new mathematical methods are being developed; it must keep in touch with this Association, to know what is being taught and how it is being taught. The Joint Council will, we hope, survey the whole field and coordinate the various activities of the constituent bodies.
- The Other Newman.
The Mathematical Gazette 54 (390) (1970), 329-335.
No history of the Christian Church in the 19th century can ignore the tremendous figure of John Henry Newman (1801-1890), striding across the scene from Calvinism through the Oxford Movement to the Cardinal's hat. Most biographers of John make some brief reference to his brother, Francis William (1805-1897), poking a little fun at his journey through various branches of nonconformity to an austere agnosticism: his eccentric pilgrimage is contrasted with the contrary progress of his brother by William Robbins in The 'Newman Brothers' (London, 1966), and is very sympathetically studied in one section of Basil Willey's 'More Nineteenth Century Studies' (London, 1956). But in neither of these books is any reference made to his considerable mathematical ability.
- Return to 1971. Shanks, Ferguson and π.
The Mathematical Gazette 55 (392) (1971), 243-248.
Fifty or sixty years ago, any forward schoolboy (to borrow a phrase from Macaulay) knew that a man called Shanks had calculated the value of π to 707 places of decimals. Who Shanks was, why he should have performed this Gargantuan feat, and why he should have stopped, if not from exhaustion, at the 707th place, were questions that may have been asked but were seldom answered. The bald statement is still to be found in a number of recent books, but it is not often brought up to date by noting the interesting sequel, that round about 1944 a member of this Association, D F Ferguson, re-calculated π by a formula similar to but different from that used by Shanks, and discovered that Shanks had made a mistake and that his value was incorrect after the 527th place. ... The Gazette for May, 1946, contained a note by D F Ferguson, then on the staff of the Britannia R.N. College, Dartmouth, with the modest title "Evaluation of π. Are Shanks' figures correct?". In May 1944, R W Morris, also of Dartmouth, had shown Ferguson a formula, new to them both, of Machin type ... which stimulated Ferguson to carry out the calculation to 100 decimal places and then, by May 1945, to 530 places, where Shanks had stopped in 1853. Values up to about this point were well established and Ferguson was able to eliminate his own occasional errors; but at this point he differed irreconcilably from Shanks. ... Ferguson checked his own figures and became convinced that Shanks had made a mistake. I was very pleased to be able to publish Ferguson's interesting discovery in the Gazette; I also suggested to him that he should write to R C Archibald, the editor of the newly established journal 'Mathematical Tables and Aids to Computation', both for the information of that periodical and for the possibility of instituting a further check. Professor Archibald suggested to Dr J W Wrench, Jr. that he might make an independent computation by means of Machin's formula; this Wrench did in collaboration with Mr L B Smith. A composite note in 'Mathematical Tables and Aids to Computation' by Archibald, Wrench and Smith, and Ferguson confirmed Ferguson's results. This note gives the value of π to 808 decimal places, also given in the Gazette for February 1948.