1.2. Review by: George Abram Miller.
Bull. Amer. Math. Soc. 29 (8) (1923), 372.

From the title of the series it is clear that the aim is to deal with fundamental theories rather than to present details. Differences of opinion naturally exist as regards what should be regarded as most fundamental. Some might regard the theory of the ∆-subgroups, which does not appear in the present volume, as more fundamental than some of the theories which do appear. In fact, some readers may not agree with a statement found on page 97 to the effect that the representation of groups by means of substitutions is the most important domain of group theory. Most readers will probably agree, however, that the material of the present volume has been, on the whole, wisely selected for the purposes in view, which seem to have included an introduction to the theories due to Frobenius. From this standpoint the present volume is especially useful.

About two years ago there appeared a small volume of 120 pages in the Sammlung Göschen, entitled Gruppentheorie, by Ludwig Baumgartner. The fact that the present volume on the same general subject appeared so soon thereafter seems to indicate that there is now a considerable demand in Germany for new introductory books along this line. This is the more remarkable since the German student was already supplied with various useful expository works, including the two books by Netto, published in 1882 and 1908, and Weber's, Lehrbuch der Algebra. While the present volume has much in common with these earlier works its main object seems to be to lead the reader rapidly into some of the more modern developments, especially into those inaugurated by Frobenius about 1896. On the whole, the volume under review furnishes a very attractive introduction into some of the most modern developments of the theory of groups of finite order, with emphasis on its applications, and we can only wish it success along with the other volumes of the interesting series to which it belongs now being published under the general editorship of R Courant of the University of Göttingen.

2.1. Review by: Philip Hall.
The Mathematical Gazette 14 (194) (1928), 148.

The second edition of Professor Speiser's book differs from the first chiefly in the presence of a new chapter, excellently illustrated, on what may be called the theory of wall-paper: as the author remarks, this is probably one of the most ancient branches of mathematics. Apart from this, the treatment of first principles has been somewhat elaborated. The author has succeeded in the astonishing feat of condensing all the principal propositions of the subject into the space of 250 pages. The style is extremely elegant and concise, with a minimum of comment and illustration. ... Altogether the author has written a most valuable book which should stimulate interest in this fascinating theory, so rich in contacts not only with other branches of mathematics but also with the latest speculations of natural philosophy.

3.1. Review by: Marshall Hall.
Bull. Amer. Math. Soc. 44 (1938), 313-314.

Although the third edition of this standard text contains only eleven pages more than the second edition, there are more than eleven pages of new material, compensation having been made by setting the type more compactly.

This work is too well known to need a detailed description. In the third edition, the main plan of the second (of 1927) has been preserved, but with the addition of a new chapter and two new sections. This has been done at the expense of only ten additional pages of text is a tribute to the author's powers of concise exposition, for there appear to be no omissions of any consequence. The present edition preserves to the full the high stylistic qualities of the earlier versions. There are many mathematical books of which one may confidently feel that every word has been weighed. But there can be few in which they have been weighed with such skill as here.

4.1. Review by: Kurt August Hirsch.
The Mathematical Gazette 42 (340) (1958), 160.

The first edition of this famous book, in the "yellow" series of mathematical monographs of the Julius Springer Verlag, is 35 years old. Further editions appeared in 1927 and 1937 and a photographic reproduction in the United States in 1945. During this long period many new branches have been added to the general theory of groups, some of which now stand in the centre of the research interests. But since Speiser's book contains some valuable chapters on aspects of group theory that cannot be found in other textbooks, the transfer to the Verlag Birkhauser is a welcome way of preserving the fine features of this classical treatment, while at the same time making room for a more modern approach.

5.1. Review by: S.
Annalen der Philosophie und philosophischen Kritik 5 (2) (1925), 70.

Speiser explains the cultural significance of mathematics in a series of 24 pieces from the literature. The development of mathematics itself, however, occurs only as a rough sketch coming out of the influence of mathematics to other areas such as the obvious religion, philosophy, art, and literature. But regardless of way it is done, it is interesting to read side by side the extracts from Plato, Aristotle, Dante, Leonardo, Kepler, Goethe, Pascal, Rousseau, etc.

"Wonderful, it seems to me, the insight which the mathematicians have gained!" Archytas of Tarentum. Such is the old-world motto which Professor Speiser of Basel puts on the title-page of his collection of chosen passages translated into German from the writings of great thinkers of all time and of many European lands. ... Moulded on the pattern of those musical albums put into the hands of the amateur, this little book has at once the charm and the weakness of an anthology. It is not what is called a mathematical book in the ordinary sense; on the other hand, it is only to one who loves and is versed in mathematics that it could appeal. We could wish to see the example set by Professor Speiser followed in our own country. It is indeed well that the mathematician should have before him in his own tongue the actual words used by great philosophers, poets and artists, expressing their views as to the nature and aims of mathematics in the various domains of our mental life.

6.1. Review by: Edward Switzer Allen.
Bull. Amer. Math. Soc. 39 (7) (1933), 484-485.

This book is not a treatise on how mathematicians think. It is a collection of essays on mathematical thought as it is revealed in art and music, in philosophy and astrology. It is the work of a man of broad culture - one whose contributions to group theory are well enough known, but who is also at home in yet more esthetic realms and is conversant with the history of serious human thought. ... [The reviewer is] urging that the book itself be read. It is not a necessity for one's library; it is a delight.

... the contents of the book are far from technical mathematical science and the scientific method and logic are not treated at all. Because such technical mathematics also needs to be constructed by its nature, its content is not completely expressible in propositions. The actual mathematical idea is due in original spirit to Plato ...

7.1. Review by: Max Dehn.
Amer. Math. Monthly 54 (7.1) (1947), 424-426.

In this book we have the philosophy of a mathematician. It is written with the enthusiasm of a distinguished mathematician who penetrates the arts and the world in his peculiar way. It will transmit, I imagine, this enthusiasm to every mathematician who is not only a craftsman but possessed by sacred fire as the poet and philosopher ought to be. ... It is interesting to realise that Speiser, in his discussions, dwells mainly on long bygone times, when the physical world and the world of the human soul were felt, by the wise men as well as by the common people, as one.

8.1. Review by: Carl B Boyer.
Isis 47 (2) (1956), 194-195.

The appearance of this book in a third edition is a token of the fact that mathematics means different things to different people. Today the majority of practicing mathematicians tend toward a formalistic view of their subject - mathematics is a meaningless game played with meaningless symbols according to certain rules agreed upon beforehand. The game takes on meaning only when it is removed from the realm of "pure mathematics" to that of "applied mathematics" by associating the symbols with things having experiential content. This view is the very antithesis of what Speiser here envisages as the mathematical habit of thought. The volume contains virtually no mathematical language, diagrams, or symbols, in the ordinary sense of the word, with the exception of a very occasional use of numbers or of letters designating quantities. There is not a single equation of the conventional sort ...

9.1. Review by: Henry Desmond Pritchard Lee.
The Classical Review 51 (6) (1937), 239-240.

The author deals only with the second part of the dialogue. A large part of the book consists of a close and accurate paraphrase of the argument, which frees it from its tiresome dialogue form. The argument is analysed into nine main positions and seventy-eight subsections. The division into subsections provides an accurate and useful guide to the dialogue. ... The commentary is often helpful, particularly when it cites mathematical analogies, but there are far more fallacies in the argument than those which the author notices, and the difficult question what we are to make of these fallacies is left untouched.

The theme of Speiser's book is the second part of the dialogue of Parmenides ...

10.1. Review by: William Hunter McCrea.
The British Journal for the Philosophy of Science 4 (14) (1953), 175-177.

Professor Speiser is a distinguished Swiss mathematician who is well qualified for the formidable task he undertakes in this book. The word 'elements' in the title is used, in a general way, with the sense it has in the title of Euclid's Elements. Speiser's object is to set out in a strictly systematic way the elements of philosophical thinking as, according to his standpoint, they have determined and still do determine the structure and content of such thinking. There is thus no claim that these elements determine anything that can be called 'absolute' laws of thought; indeed, Speiser evidently expects that in course of time they will be superseded by some other, 'non-Platonic', system.

A prelude and a fugue, separated by a chapter devoted to remarks on the former and preparation for the latter, give the basic elements of Hegel's philosophy of science, with the critical comments and the modifications, which are dictated by mathematical considerations. The author's profound understanding of classical and of modern philosophy make it possible for him to indicate the essential defects in Hegel's philosophy. Discrete and continuous, being and not-being, growth and decay, the thesis, antithesis and synthesis of the Hegel dialectics come in for thorough discussion. This book will be of interest chiefly to mathematicians with a taste for and a good training in philosophy, well acquainted with Hegel's writings.

In the introduction Professor Speiser describes his book as finger exercises for beginners in philosophical and mathematical thought. Fundamental concepts in philosophy and mathematics are subjected to the Hegelian dialectic to bring out their inner content. ... The poverty of Hegel's own contribution to the foundations of mathematics was due, Speiser considers, to the lack of interest in mathematics in Germany at the time; France's Lagrange, Laplace, Legendre, Carnot, Dupin, Lamé, Monge, Poncelet, Lacroix and Cauchy were challenged in Germany only (but outstandingly) by Gauss. ... Speiser makes a number of interesting observations on the great mathematicians philosophising about their subject, and draws some striking parallels between mathematics and musical composition (the book in fact is written in fugue form) but the major part of the work is totally alien to English philosophical thought, and has no point of contact with present trends in the foundations of mathematics.

11.1. Review by: Martin Hugo Löb.
The Mathematical Gazette 41 (338) (1957), 310.

The book consists of a collection of lectures and essays ranging over diverse topics in art, science, philosophy and theology. Several of these had previously been published independently. Their underlying thesis is that most important intellectual and artistic productions are intrinsically organised in accordance with some mathematical structure which must be consciously understood for their proper appreciation. Mathematics, science and art form an essential unity, and a strict division between them can be observed only at a loss to each of them. The Renaissance painters, for instance - themselves mathematicians - incorporated in their work the then most modern results in geometry. Art has fallen into decline since its development was outstripped by mathematical progress. The main thesis is developed by analysing a number of ingeniously chosen examples, rather than by a systematic exposition of the author's ideas. The importance of group theory is stressed, but surprisingly, perhaps, in book which deals extensively with mathematical thinking, little more than incidental references are to be found regarding foundational research in mathematics later than Fichte. The author claims that mathematics requires a platonistic foundation and that mathematical truth is absolute.