**1. The Theory of Substitutions and its Applications to Algebra (1892), by Eugen Netto.**

**1.1. Review by: Oskar Bolza.**

*Amer. Math. Soc*.

**2**(5) (1893), 83-106.

Netto's "Substitutionentheorie und ihre Anwendungen auf die Algebra" appeared for the first time in 1882; it was followed, in 1885, by an Italian edition, and now we have the pleasure of welcoming an English edition, revised by the author and translated into English by Dr Cole. The mathematical public at large, and the English-speaking part of it in particular, are greatly indebted to Dr Cole for his careful and expert translation. Mastering the subject as well as both languages in full extent, Dr Cole has transformed the sometimes rather tough material into clear and fluent English. We are especially obliged to him for the fortunate choice of many technical terms, alien so far to the English mathematical language. We are equally indebted to the author for the numerous valuable additions by which this new edition has been enlarged and improved. The great merit of Netto's book consists in the skilful and highly pedagogical presentation of the theory of substitutions, given in the first part of the book. The reader is gradually led from the most elementary considerations on symmetric and alternating functions to the general theory of unsymmetric functions of *n *independent elements, out of which the theory of substitutions is step by step evolved, the unsymmetric functions serving all the while as a concrete substratum for the abstract conclusions of the theory of substitutions. By this means an easy and attractive entrance into the theory of substitutions is gained, accessible even to the beginner, and it may fairly be said that Netto's book has largely contributed to spread the knowledge of this important branch of mathematics.

**2. Lehrbuch der Combinatorik (1901), by Eugen Netto.**

**2.1. Review by: Florian Cajori.**

*Science, New Series*

**16**(403) (1902), 469-470.

At the present time neither European nor American universities offer lecture courses on the subject of combinatorial analysis. This fact is the more noteworthy when we remember that during the first quarter of the nineteenth century nearly every mathematical chair in Germany was occupied by a specialist in that field. This Combinatorial School of Germany has passed into deserved oblivion. Under the leadership of C F Hindenburg it represents the culmination of an unfortunate tendency of eighteenth century mathematicians to develop analysis, particularly the subject of infinite series, with reference to form only, and to pay little or no attention to the actual contents of formulae. The polynomial theorem was hailed as 'the most important theorem of all analysis.' In combinatorial analysis (combinatoric) the German school was contented with the deduction of rules for the writing down of all the combinations and permutations that are possible under given restrictions. The simple fact that the able and fairly complete treatise now under review hardly mentions the work of Hindenburg shows that what are now considered the substantial parts of combinatoric have been developed outside of the German Combinatorial School. Associated with the early development are the great names of Pascal, Leibnitz, Wallis, James Bernoulli and De Moivre. While combinatoric is not now made the subject of lectures in our universities, it is nevertheless of importance. The student acquires much of it during the pursuit of other branches. It is touched upon in the study of ordinary algebra, of determinants, of substitution and group theory, of the theory of numbers, and of the theory of probability. Netto's book is of value as a reference book, especially as no text of importance on combinatoric has been published for sixty-five years. In arrangement and selection of material it resembles somewhat Netto's brief article 'Kombinatorik' in the 'Encyklopidie der Mathematischen Wissenschaften.' The book takes notice of researches of recent date, including several papers by American authors. Starting out with the fundamental definitions the author treats of combinations, permutations, and variations under different limiting conditions, leading up to various problems, as, for instance, Tait's problem of knots. Combinations and variations are considered under the restriction of a definite sum or a definite product of the elements. The partition of numbers and Durfee's graphs are taken up. In the course of further combinatorial operations the author studies systems of triads arising in connection with Kirkmann's and Steiner's problems. Steiner's queries have not yet been fully answered. Kirkmann's is the 'Fifteen School Girl Problem': 'To walk out fifteen girls by threes, daily for a week, without ever having the same two together.' In the discussion of this it is to be regretted that Netto overlooked E W Davis's pretty 'geometric picture,' given in the *Annals of Mathematics*, Vol. XI., 1897, where a one-to-one correspondence is established between the fifteen girls and fifteen points on a cube; eight points at the corners, six at the mid-points of the faces, one at the cube-centre; the thirty-five triads are then easily found. Netto's book is substantial food for the average reader. Yet some topics in combinatoric were originally suggested by questions propounded for amusement. The 'problem of the eight queens' is of this nature. Eight queens are to be placed upon a chessboard so that none of them can capture any other. It was first propounded in Berlin in 1848 and has 92 solutions.

**2.2. Review by: Anon.**

*The Mathematical Gazette* **2** (35) (1902), 216-217.

One of the latest additions to the excellent series of monographs published in the *Sammlung von Lehrbucher auf dem Gebiete der mathematischen Wissenschaften* is from the pen of Professor Netto of the University of Giessen, whose name is familiar in connection with the substitution theory. It is a common complaint among students that in questions on what Prebendary Whitworth so aptly called "Choice and Chance," that they never know exactly where they are. Although much of the glorious uncertainty which lingers in their minds after attempting the solution of a problem may be almost entirely due to careless or ambiguous wording, one cannot help feeling that if the student is left in the air with regard to these and kindred problems it is because of the cursory treatment which this branch of mathematics receives. The interest of the subject is practically inexhaustible. Such a volume as this draws on arithmetic, algebra, analysis, and probabilities for its material; the questions which are set are some of great historical interest; some are as amusing as they are exasperating, and almost all of them require a clear brain and a judicial temperament

**3. Lehrbuch der Combinatorik 2. Auflage (erweitert und mit Anmerkungen versehen von Viggo Brun und Th Skolem) (1927), by Eugen Netto.**

**3.1. Review by: Francis Puryer White.**

*The Mathematical Gazette*

**15**(206) (1930), 82-83.

This is a reproduction by "Photomechanisches Gummidruckverfahren" of the first edition, published in 1901, with the addition of notes and of two fresh chapters. Since the first edition was fully reviewed in the Mathematical Gazette (October 1902), it is only necessary to comment on the new matter. The notes (pp. 309-338), by T Skolem, of Oslo, are devoted partly to simplifications of proofs given in the text and partly to recent developments. The appearance of P A MacMahon's *Combinatory Analysis* (2 volumes, Cambridge, 1915-6) is naturally of great importance in the latter respect; in particular we may notice that Netto's remark (p. 74) that the treatment of Latin squares has not yet left "das Stadium der Spielerei" is no longer justified in view of MacMahon's investigations. Another matter in which progress has been made is in the generalisation of the eight queens problem; here the appropriate reference is to the later editions of Ahrens: *Mathematische Unterhaltungen und Spiele*, especially to Polya's work, therein contained, on the so-called doubly-periodic solutions, in which the chess-board is supposed to be infinite and the problem is so to arrange the queens that any square board, of side n, taken out of the whole, contains n queens which cannot take one another. The two additional chapters are by Viggo Brun and Th Skolem respectively. The first deals with the "distribution-function," defined rather more generally than that which MacMahon introduced with so much skill and profit. ... Skolem's chapter is devoted to groupings of different objects into systems, which may have objects in common; for example, Steiner's arrangement of 7 symbols in threes such that any two occur together in one and only one three. ... The book is well produced ... One would have been greatly interested to have read Major MacMahon's comments on this second edition; the book was in fact sent to him for review on its appearance, and it is only his illness and recent death, on Christmas Day, 1929, that has deprived readers of the Gazette of an authoritative opinion from the unrivalled master.

**4. Elementare Algebra (1904), by Eugen Netto.**

**4.1. Review by: John Henry Tanner.**

*Bull. Amer. Math. Soc*.

**17**(10) (1911), 547-548.

The book under review is written in the informal and somewhat detailed style of the lecture as distinguished from a treatise, is rather generously supplied with well-chosen figures, and contains a good index but no exercises for the student to solve. It is the outgrowth, as the author tells us in the preface, of a course of lectures (entitled *Einleitung in die Algebra*) which he gives during the summer semester of each year in the University of Giessen. The purpose of the book, and also of the lectures on which it is based, is two-fold : for those students who are to continue their work in mathematics, it is designed to bridge over the gap which usually exists between the algebra of the fitting school and that of the university; and to the non-mathematical student it presents, in a somewhat popular form but from a broad viewpoint, some of the more important problems and methods of algebra, a knowledge of which should have a place in a liberal education. Equations of the first four degrees determine the main divisions of the book, and around these cluster a great variety of topics. ... The book as a whole is carefully planned and well written ; it broadens the student's point of view, stimulates his interest in the subject, and gives him no false notions which he will have to unlearn later. In the opinion of the reviewer it will prove itself very helpful, not only to the class of students for whom it was especially prepared, but also to teachers who are engaged in this field of work ; it is especially to be commended to the attention of mathematics teachers in our own secondary schools.

**5. Gruppen- und Substitutionentheorie (1908), by Eugen Netto.**

**5.1. Review by: Harold Hilton.**

*The Mathematical Gazette*

**4**(76) (1908), 397-398.

A few years back a reviewer pointed out in these columns the need of a really elementary Introduction to the Theory of Groups. Attempts have since made to satisfy this need, among which is included Dr Netto's treatise. The author has succeeded in producing a book which is elementary and is quite the sort of thing a beginner wants. The exposition is very clear, and the earlier chapters contain many useful illustrative examples. These unfortunately grow scarcer as the book proceeds, and examples are nowhere provided for the student to test his own skill. The scope of the treatise includes the theory of composition-series, chief-series, Sylow subgroups, and Abelian groups, but not the theory of commutants or groups of isomorphisms. A considerable amount of space is devoted to soluble groups. It is a pity that the list of results on p. 122 is inserted, since many of them have been rendered obsolete by Burnside's proof that every group of order *p*^{α}*q*^{β} is soluble - a result not included in Dr Netto's list. Considering the elemenltary nature of the book, the treatment of permutation-groups in chaps. 9, 10, 11 is very thorough; and will prove useful even to more advanced students. To stimulate interest the author quotes many results of which proofs are omitted for lack of space. The idea is a good one, but its value is largely neutralised by the fact that no references are given to places where proofs may be found. This seems a mistake, even if it is freely admitted that references in the case of proved results are out of place in an elementary treatise. The book, though good as far as it goes, needs supplementing, for it is impossible nowadays to study group-theory satisfactorily without some acquaintance with the substitution-groups which promise to play a more important part even than permutation-groups in future developments of the subject.

**5.2. Review by: William Benjamin Fite.**

*Bull. Amer. Math. Soc*. **16** (1) (1909), 33-35.

In conformity with the general plan of the Sammlung Schubert Professor Netto aims to give in this book an introduction to the theory of groups of finite order. He has succeeded admirably in his purpose. Those readers who are not already familiar with the details of the theory will find Chapter II particularly valuable in fixing for them the fundamental notions of the subject, if they take the pains to work through the details. Indeed we do not know if there is another place where this particular phase of the subject is treated so happily. But this is by no means the only good chapter. They are all excellent and the book as a whole is a fine example of clear and attractive exposition. The author introduces some new notation which is of value in the interest of brevity. ... In publishing so excellent a treatment of the subject Professor Netto has performed a service of value to the mathematical public.

**6. Die Determinanten (1910), by Eugen Netto.**

**6.1. Review by: James Byrnie Shaw.**

*Bull. Amer. Math. Soc*.

**17**(10) (1911), 547-548.

The enormous growth in recent years of both mathematics and engineering is nowhere shown better than in the appearance in both Europe and America of many small handbooks particularly designed to give in little space the main features of some definite field of these large subjects. This volume is of such a character. It undertakes to develop the elementary theory of determinants with some applications. The applications are not technical. The first two chapters are devoted to the elementary properties and expansions. Then follows one on evaluation, and one on products. A chapter follows on arrays, which are here called matrices. Then comes a chapter on particular determinants, one on the solution of linear equations, and one on resultants. Following is a chapter on linear substitutions, which could have properly formed part of the chapter on matrices. In our opinion, it is a mistake to consider the array of symbols merely as an array. The composition of matrices, if not the mere ordering, makes the symbols stand for more than mere arrays. They are indeed *n*^{2}-fold multiple quantities and should at least be treated as multiplexes if not as operators. From this point of view the linear substitution is merely one example of a matrix. A chapter is devoted to geometric applications. The book is closed by a very brief chapter on differentiation and one on functional determinants, which last however would have been labeled more correctly "The Jacobian." As a whole the treatment is clear, well illustrated, and all that could have been given in the space.