Review: An Elementary Treatise on the Differential Calculus, with applications and numerous examples, by Joseph Edwards.

When a mathematical text book reaches a second edition, so much enlarged as this, we know at once that the book has been received with some favour, and we are prepared to find that it has many merits. We are at once struck by Mr Edwards' lucid and incisive style; his expositions are singularly clear, his words well chosen, his sentences well balanced. In the text of the book we meet with various useful results, notably in the chapter on "some well known curves" and moreover the arrangement is such that these results are easy to find; and in addition to these, numbers of theorems are given among the examples, and, this being a feature for which we are specially grateful, in nearly every case the authority is cited. Recognizing these merits, however, we notice that the book has many defects, some proper to itself, some characteristic of its species; and just because it is so attractive in appearance, it seems worth while examining it in detail, and pointing out certain specially vicious features. A book of this size may fairly be required to serve as a preparation for the function theory; at all events, the influence of recent Continental researches should be evident to the eyes of the discerning. Mr Edwards' preface strengthens this reasonable expectation, for he promises us " as succinct an account as possible of the most important results and methods which are up to the present time known." But we soon find that the "important results and methods" are those of the Mathematical Tripos; and in our disappointment we utter a fervent wish that instead of the "large number of university and college examination papers, set in Oxford, Cambridge, London, and elsewhere" Mr Edwards had consulted an equally large number of mathematical memoirs published, principally, elsewhere. The Mathematical Tripos for any given year is not intended for a Jahrbuch of the progress of mathematics during the past year; and as long as so many will insist on regarding it in that light, text books of this type will continue to be published. ... Lejeune Dirichlet's definition of a function is adopted. ... That Mr Edwards does not adhere to this definition is evident from his tacit assumption that every function can be represented by a succession of continuous arcs of curves. ... No more damaging charge can be brought against any treatise laying claim to thoroughness than that of recklessness in the use of infinite series; and yet Mr Edwards has everywhere laid himself open to this charge. One of the most difficult things to teach the beginner in mathematics is to give proper attention to the convergency of the series dealt with. All the more need, then, that a text book of this nature should set an example of consistent, even aggressive carefulness in this respect. ... We find no formal recognition of the importance of uniform convergence in modern analysis, nothing even to suggest that he has ever heard of the distinction between uniform and non-uniform convergence. We begin to suspect that he has never looked into Chrystal's Algebra. ... Feeling now somewhat familiar with Mr Edwards' point of view, we examine his proofs of the ordinary expansions with a tolerably clear idea of what we are to expect. We find, of course, "the time-honoured short proof of the existence of the exponential limit, which proof is half the real proof plus a suggestio falsi"; we find in the chapter on expansions a general disregard of convergency considerations ... We pass on now to the second part, applications to plane curves; and here we must object emphatically to the introduction of so many detached and disconnected propositions relating to the theory of higher plane curves. ... The whole exposition of this theory of expansion is most inadequate. ... We object, then, to Mr. Edwards' treatise on the Differential Calculus because in it, notwithstanding a specious show of rigour, he repeats old errors and faulty methods of proof, and introduces new errors; and because its tendency is to encourage the practice of cramming "short proofs" and detached propositions for examination purposes.

Review: Introductory Modern Geometry of Point, Ray, and Circle, by William Benjamin Smith.

In reviewing a book, one of the canons of fair criticism is to regard its adaptation to the readers for whom the author himself designs it; but as a preliminary to this notice, we must object to the selection implied in the preface, where Professor Smith describes his book as intended "to present in simple and intelligible form a body of geometric doctrine acquaintance with which may fairly be demanded of candidates for the Freshman class," and then points out that one year's study of geometry is about as much as can be expected in schools. Our own conviction is that geometry may with great advantage be taught, to children in their early school days. The simplest kind of geometry, of course; with few formal proofs, and depending more on the teacher than the text-book. But even when this early introduction has been omitted, the subject seems to be one that may be presented in formal guise to the average child of twelve or thirteen. It relates to that with which he is already practically familiar, illustrations may be drawn from his every-day experience, his unconscious perceptions of space relations may be appealed to and formulated; and thus it presents itself as a valuable discipline by which his reasoning faculties may be developed, and his vague disconnected perceptions organized, without burdening him with a mass of new and possibly uninteresting facts.

But just because geometry is so eminently fitted for the youthful mind it should be at first presented in such form as to be in accord with the general views of laymen, when these are not in direct opposition to the truth. It may well be that the ideas of modern geometry on such questions as that of the nature of space ought to be explained earlier and to more students than is now done; but if we accept W K Clifford's rule, "before teaching any doctrine, wait until the nature of the evidence for it can be understood," such discussions will not be put in the forefront of our geometrical teaching. They will be more easily and profitably treated when they can be founded on knowledge derived from a study of Euclidean space. "There is no time of reading a book better than when you need it, and when you are on the point of finding it out yourself if you were able," says J Clerk Maxwell; why then should we thrust upon the student "the consoling hope that, after all, this other [view of space] may be the true state of things," until he is able to appreciate the relief thus offered, through having, with Clifford, suffered from " the dreary infinities of homaloidal space"? But given this preliminary study of geometry, then comes the time for a concise systematic treatment of the subject with direct reference to modern ideas. Certainly all teachers of even the most elementary geometry ought to know wherein these modern views differ from the older ones; and to those that have neither opportunity nor inclination for abstruse mathematical studies, a book on the lines of the one before us will afford valuable assistance. To some such circle of readers, therefore, we prefer to regard this introductory modern geometry as addressed.

... the book, with all its good points, is hopelessly marred by the author's persistent disregard of conventional nomenclature. If he had entirely invented the science, or at any rate this special development of it, we could not deny him the legal right of naming his own creations, even though we might deplore his unfortunate choice of such terms as tract (finite straight line), perigon (four right angles), numeric, used as a noun, and finity, in contradistinction to infinity. But all the ideas here set forth are the common property of the mathematical world; the science of geometry was not invented yesterday, and it is already provided with a fairly complete and satisfactory English vocabulary. It requires very cogent reasons to justify a scientific writer in deviating from established usage in nomenclature or notation. Why speak of the base circle in reciprocation as the referee? Why speak of the point of tangence rather than of the point of contact? Why, in short, disfigure the work with the multitude of strange and uncouth terms that we find here? Have not the most effectual reformers been conservative in trifles?

Review: Vortrâge über ausgewàhlte Fragen der Elementargeometrie, by Felix Klein.

Thus in these few pages some of the most striking results of modern mathematics are made accessible to many who would otherwise hardly have heard of them. But while reading this brilliant exposition it is difficult to avoid cherishing a lurking regret, which is possibly very ungracious, that Klein could not himself spare time to arrange his work for publication; for though we have here in full measure the incisive thought and cultured presentation which together make even strict logic seem intuitive, yet at times we miss the minute finish and careful proportion of parts that we feel justified in expecting from him. And yet revision and consolidation might have seriously interfered with the graphic simplicity of these chapters, and left them less adapted to their special purpose.

Review: Julius Plücker's gesammelte mathematische Abhandlunge, edited by Arthur Schönflies.

The volume whose separate title is given above is the first of the two which together form this collected edition. The only point in the arrangement that could be wished changed is the natural decision, referred to in the preface, not to reprint any of Plücker's longer works, inasmuch as they are still to be bought in the original editions. But interesting as these original editions may be, they are not satisfactory, for type and paper alike leave much to be desired; it is hardly possible, for instance, to read the Theorie der algebraischen Curven with any enjoyment; and so it is perhaps permissible to express the hope that at some future time, not too far distant, these also may appear in as readable a form as the Collected Papers.

Report: On the International Congress of Mathematicians at Paris in 1900.

One thing very forcibly impressed on the listener is that the presentation of papers is usually shockingly bad. Presumably the reader desires to be heard and understood; to compass these ends, instead of speaking to the audience, he reads his paper to himself in a monotone that is sometimes hurried, sometimes hesitating, and frequently bored. He does not even take pains to pronounce his own language clearly, but slurs or exaggerates its characteristics, so that he is often both tedious and incomprehensible. These failings are not confined to any one nationality; on the whole the Italians, with their clear and spirited enunciation, come nearest to being free from them. It would be invidious and impertinent to mention names; the special sinners sit in both high and low places. But it is perhaps pardonable to refer to M Mittag-Leffler's presentation of his paper to Section II as showing in how admirable and engaging a style the thing can be done. It is not given to everyone to do it with this charm; but there is not excuse for any normally constituted human being, sufficiently versed in mathematics, failing to interest a suitable audience for a reasonable time in that which interests himself, always provided that it be of sufficient novelty either in matter or in mode of treatment to justify him in presenting it at all.