The formulæ for the powers and the reversion of any given series are not so readily accessible to those who are dependent upon modern mathematical text-books as they should be considering the frequent and manifold uses to which such formulæ can be put. I thought therefore that the following compendium of such formulae would be useful to readers of the Mathematical Gazette. The chief formulae in this paper are given in De Morgan’s Differential and Integral Calculus, but this book must be getting increasingly unavailable to ordinary readers except in reference libraries.

In ordinary arithmetic a number ab … jk represents the sum of the place values of its digits a, b, …, j, k, these place values being k, 10j, 102i, etc. Also the representation is unique if the digits are restricted to being positive and less than 10. We are going to consider the representation of whole numbers by means of digits with place values, but with a different definition of place value. The digits a, b, …, j, k + 1 of a given number A are to be positive, and the number of them, n, is to be assigned, the units digit k + 1 (in the standard form) is to be greater than zero, and the place value of the digit a in the nth place counting from k + 1 is to be a. a + 1 … a + n - 1 divided by 1.2. …. n, or the number of homogeneous products of a letters taken n at a time; this number or place value is written (a)n.

In the present month Professor G. H. Hardy, who for eleven years has been resident in Oxford, will return to Cambridge and occupy the Sadleirian chair of mathematics, vacant by the resignation of Professor Hobson. The occasion seems a favourable one to recall the history of the Sadleirian chair and of the three distinguished living mathematicians who have occupied it. All of them have been Presidents of the Mathematical Association, and have exercised a great influence upon the teaching of our subject. We shall deal at some length with this aspect of their work. Some account will also be given of their researches, but although from a higher standpoint this is the most important of their activities, an adequate estimate of their original work will not be attempted on this occasion.

Dr. Robb, in his rejoinder to my criticism of his paper, asks me a direct question; and as, to me at least, the matter under discussion is of far greater importance than any mere difference of opinion between us, I hope I may be permitted to reply.

The subject of this paper occupies a strange position among the fundamental concepts of mathematics. The beginner uses limits with equal freedom, or, as some would say, with equal recklessness, in analysis and geometry, but whereas the time comes when the analytical limit is submitted to the severest scrutiny, the examination of the geometrical limit forms no part of any university course and comes into no treatise. There is not even an essay on the subject in the collection of Questioni riguardanti la Geometria elementare, edited by Enriques, or a section dealing with it in Klein’s Elementarmathematik vom höhere Standpunkte aus. When his turn comes to teach, every mathematician knows all about limits in analysis, is familiar with half a dozen ways of looking at them, and indeed has a basis of thorough understanding from which to exercise his discretion as to the treatment proper for one class of students or another. Of limits in geometry, except perhaps in three special cases, he has probably never heard a critical word.

1. The method given by Euler * for the transformation of series was used by him to obtain “sums” for various divergent series, and recently Knopp † 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 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.‡ De Morgan § 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 a t 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.

Nobody who has paid attention to the acknowledgments of gifts to the Library during the past seven years will have failed to observe that Mr. Greenstreet was the most lavish and the most persistent of benefactors. The records of the books of which he, as he put it, unloaded his own shelves as soon as the Library was removed to Reading, occupy several pages of vols, xi and xii of the Gazette; his name is in many of the volumes which were in the Library before that time, and few months have passed since without evidence of his goodwill.