**1. Die Lehre von den Kettenbrüchen (1913) by Oskar Perron.**

**1.1. Review by: Joseph Lawrence Nayler.**

*The Mathematical Gazette*

**7**(106) (1913), 159-160.

In writing his work on Continued Fractions, Dr Oskar Perron has filled a gap in mathematical literature at a place where for some time a work has been needed. The original papers on the subject are now very numerous, and work a very good selection of available material has been made. Great trouble seems to have been taken to give a very clear formulation of the theorems proved and to set down the proofs in a lucid manner; the rigidity of modern analysis is also observed in most parts of the book. ... As a work of reference the book will be of extreme value to the more advanced student, who will find at the end a lengthy and useful index of original papers on the subject.

**2. Die Lehre von den Kettenbrüchen (Second edition, 1950) by Oskar Perron.**

**2.1. Review by: Thomas Arthur Alan Broadbent.**

*The Mathematical Gazette*

**36**(315) (1952), 66.

The first edition of Perron's classic appeared in 1913, the second, substantially a photographic reprint with some corrections and additions, in 1929; the Chelsea Company of New York has now produced a reprint in its usual neat and pleasing format. The book is an indispensable work of reference for the library, and for the shelves of the working mathematician whose interests are analytical. Its two sections, on the arithmetic theory and the analytic theory, give a comprehensive account of the topics and omit nothing of importance from what was known in 1913. Later special studies, many of them due to American mathematicians, are to be found in Wall's recent book, but in a field where the textbook literature is not extensive, Perron remains the best guide for the novice. The style is simple and precise and presents no difficulties to a reader having a firm grasp of the fundamental principles of elementary analysis. In adding Perron's masterly treatise to their list of reprints, the Chelsea Company have once more put us in their debt by making available a book without which no mathematical library is complete.

**3. Die Lehre von den Kettenbrüchen (Third edition). Vol. I: Elementare Kettenbrüche (1954) by Oskar Perron.**

**3.1. Review by: George Neville Watson.**

*The Mathematical Gazette*

**41**(338) (1957), 309-310.

In the first edition of Perron's famous book (published in 1912 and reviewed in Vol. 7 of the Gazette) the author was able to assert with truth that no comprehensive exposition of the subject of continued fractions was in existence; it is due to him that the assertion thereupon ceased to be true. Since the day that the book first saw the light it has been not only a fascinating book for the mathematician to read, but also one of the predestined members of the select class of works which are rightly described as mathematical classics. The book reached a second edition in 1929, a reproduction of the first edition with some very minor changes and additions to bring it up to date; and now, quarter of a century later, we can greet the inspiring friend and companion of some forty years, wearing a new frock but still readily recognisable. The volume now under review consists of a revised version of the first five chapters, whose subject can be described as elementary continued fractions; the width of the page has been increased by about one-tenth and a somewhat more concise style has been adopted in places. ... This volume makes the reviewer look forward eagerly to experiencing the thrills which will be given by a sight of the more advanced and more interesting parts of the subject to be contained in the second volume of this new edition.

**3.2. Review by: Hubert Stanley Wall.**

*Mathematical Reviews* MR0064172 **(16,239e).**

This is vol. I of a two-volume third edition of the author's book on continued fractions, based upon Part I (the elementary arithmetic theory) of the earlier editions [2d ed., Teubner, Leipzig, 1929]. The material has been rewritten, largely in the spirit of the earlier editions, with improvements and five new sections.

**3.3. Review by: W T Scott.**

*Bull. Amer. Math. Soc.* **61** (6) (1955), 594.

Earlier editions of 'Die Lehre von den Kettenbruchen' are divided into two parts, the first containing the arithmetic theory of continued fractions and the second containing the analytic theory. The author's plan for the third edition is to treat these two parts in separate volumes. Volume I is an enlargement and improvement of what was previously Part I.

**4. Die Lehre von den Kettenbrüchen (Third edition). Vol. II: Analytisch-funktionentheoretische Kettenbruche. (1957) by Oskar Perron**

**4.1. Review by: W T Scott.**

*Bull. Amer. Math. Soc.*

**64**(5) (1958), 299-300.

Earlier editions of this text on continued fractions contained in a single volume a part devoted to the arithmetic theory and a part devoted to f unction-theoretic aspects (analytic theory). In the third edition these two parts have been published in separate volumes. Volume I was published in 1954 and the publication of the present book, Volume II, marks the completion of the third edition. ... The stated objective of the book is to give in an easily intelligible way the present state of knowledge of the subject. The author has been confronted with the difficult task of selecting and coordinating the material of major importance and not all readers will agree with his selections. Any defects of the book are those of omission. The reviewer regrets the omission of the methods and viewpoint of positive definite continued fractions and, in particular, positive definite J-fractions. However, the numerous virtues of the book, among which are clarity of presentation, systematic citing of origins of theorems, and the many examples and formulas, will make it a valuable reference for many years to come.

**4.2. Review by: Evelyn Frank.**

*Mathematical Reviews*** **MR0085349 **(19,25c).**

This is volume II of the third edition of this book, the first edition of which appeared in 1913 and the second edition with a few changes in 1929 (Leipzig). The present volume, as was the case with volume I [1954], has been very materially changed and brought up to date. It deals entirely with the analytic, function-theoretic aspect of continued fraction theory, whereas volume I deals entirely with the elementary arithmetic part. ... In all, he has given a fine account of the theory of continued fractions, all of which is of both practical and theoretical interest.

**5. Irrationalzahlen (1921), by Oskar Perron.**

**5.1. Review by: Aubrey John Kempner.**

*Bull. Amer. Math. Soc.*

**29**(1) (1923), 34-36.

The volume under consideration contains two parts of essentially different character. While the first half may be said to give a systematic treatment of the notion of an irrational number and its historical development, the last chapters are devoted to the interesting and not very widely known subject of the various methods of representing irrational numbers and their approximation by rational numbers and to a brief discussion of a certain class of transcendental numbers. These last chapters, while not containing a large amount of new material, offer much that may be of interest even to professional mathematicians, particularly since the problem of approximation of irrational numbers is steadily gaining in importance and represents a difficult, but fascinating, field for research in which German and English mathematicians are making remarkable discoveries.

**6. Irrationalzahlen (Second edition, 1939) by Oskar Perron.**

**6.1. Review by: Joseph Langley Burchnall.**

*The Mathematical Gazette*

**24**(259) (1940), 137-138.

This book develops in great detail the theory of sections with the corresponding theory of bounds, limits, powers and logarithms. A natural sequence of ideas leads to a discussion of the expressions of numbers as decimals and continued fractions as well as in the less familiar forms of Cantor's series and product and the series of Sylvester, Luröth and Engel. The criterion for irrationality is given in each case. An irrational number may be either algebraic or transcendent; since algebraic numbers are enumerable, transcendent numbers must exist. The final chapter discusses their construction with special reference to Liouville's numbers and the criteria for them. The work of Minkowski and others on criteria for algebraic numbers is mentioned but not discussed and the chapter concludes with proofs of the transcendence of e and π. The longest section of the book deals with the problem of approximating to irrationals by rationals. ... Apart from a substantial increase in the bibliography, this section is the only addition of importance and those who possess the first edition may rest well content therewith. To others the book may be heartily recommended, not only as a handy work of reference where we may find how to do the things we know can be done but are too lazy to do, but as a useful introduction to interesting and important problems.

**6.2. Review by: Olive Clio Hazlett.**

*Bull. Amer. Math. Soc.* **46** (1) (1940), 15.

As the author says in his interesting preface to this second edition, there are no great changes except in the fifth chapter, which is longer by twelve pages and includes Estermann's beautiful proof of Kronecker's approximation theorem and a section on the *Gleichverteilung. ... * The preface of this new edition seemed in some ways the most interesting part, on account of the reason that the author gives for presenting Dedekind's theory of irrational numbers rather than the theory of Cantor and Méray. In 1921, he did not seem to think it necessary to give any reason for basing his treatment on Dedekind's work; but in 1939, he devotes most of the preface to justify his giving Dedekind's rather than Cantor's theory. He refers to an article by Bieberbach and the famous one by Hardy in "Nature" on the J-type and S-type of mathematicians. One could easily wonder just what lies behind these careful justifications. However that may be, we wish him well, for Perron has done yeoman service in writing textbooks for universities and *technische Hochschulen.*

**7. Irrationalzahlen (Second edition reprint, 1951) by Oskar Perron.**

**7.1. Review by: G L Watson.**

*Biometrika*

**44**(1/2) (1957), 299.

This is a reprint of the second edition of Perron's well-known work. (The third edition, of 1947, differed from the second only in one page and a footnote.) The paper and printing are good, and the reviewer could find no serious misprint. Starting from a set of twenty-one axioms satisfied by the rational numbers, the author defines irrational numbers as Dedekind sections of the rationals, and shows that they satisfy the same axioms. He goes on to deal with limits, powers and logarithms. Then without further use of analysis he discusses various methods (particularly continued fractions) of approximating to irrational by rational numbers. The degree of accuracy attainable is investigated. He concludes with a chapter on algebraic and transcendental numbers, proving that the numbers e and π are transcendental. This means that neither of them is a root of an algebraic equation, of any degree, with integral coefficients. The work may be regarded largely as an introduction to the author's *Die Lehre von den Kettenbrüchen*. There is a good bibliography.

**7.2. Review by: Lee Albert Rubel.**

*American Scientist* **45** (4) (1957), 298A, 300A.

0skar Perron's *Irrationalzahlen* is a careful and systematic treatise on the subject that enters deeply into the theory only briefly, giving most of its space to a thorough presentation of the preliminaries. The first four chapters provide a development of the real number system by means of Dedekind cuts of rationals, a development of the elementary facts about the limit process, a careful exposition on the exponential and logarithmic functions, and a presentation of the various ways of representing real numbers, as by infinite decimals, continued fractions, and so on. The final two chapters deal in turn with the approximation of irrational numbers by rational numbers, and with algebraic and transcendental numbers. It is in the last chapter that e and π are proven transcendental. Except for the last proof in the book, no training beyond high school mathematics is prerequisite. Nor is this a book that makes heavy demands on "mathematical maturity"; rather, it is one that gently nurtures it.

**8. Irrationalzahlen (Fourth edition, 1960) by Oskar Perron.**

**8.1. Review by: Jan Popken.**

*Mathematical Reviews*MR0115985

**(22 #6782).**

The changes in this new edition of this well-known book are mainly confined to the first three chapters, where Dedekind's theory for the foundation of real numbers and related topics are treated. The general plan is unchanged, but there are several improvements which make this part still more readable. The remainder of the book, however, shows only minor alterations and additions. The fourth chapter contains not only an excellent introduction to the theory of continued fractions, but also the representation of real numbers by the series of Cantor, Lüroth, Engél and Sylvester. As such it still stands apart from all other elementary textbooks I know. The remaining two chapters contain short introductions to diophantine approximations and to transcendental numbers. The treatment of these subjects is masterly, but the reviewer deplores that these chapters - in view of important recent additions to these theories - have not been worked out somewhat more.

**9. Algebra (2 volumes) Vol 1 Die Grundlagen, Vol 2 Theorie der algebraischen Gleichungen (1927), by Oskar Perron.**

**9.1. Review by: Arthur B Coble.**

*Amer. Math. Monthly*

**36**(4) (1929), 224-225.

It is the express purpose of the author to write a book which, though designed primarily to meet the needs of advanced students, shall also be of service the investigator. This double aim has been admirably accomplished for the book is more than a transcription, with pedagogic alterations or the more common pedagogic deletions, of existing treatises. A far share of the important duty of putting modern mathematical research into connected exposition has been skillfully done. In the view of the author algebra is that part of analysis which is based on the rational operations rather than on the notions of relative magnitude and of limit. The only important departures from this limitation are those inevitably involved in the proof of the fundamental theorem that every equation with coefficients in the field of complex numbers has a root belonging to that field, and in the numerical calculation of the roots. The notion of field is introduced at the outset and theorems are stated thereafter with reference to their field of validity. The student of the subject will be attracted by the orderly and careful development, and by the sympathetic guidance afforded by the illuminating comment at various stages of progress. The historical references are adequate though not especially numerous.

**9.2. Review by: Olive Clio Hazlett.**

*Bull. Amer. Math. Soc* **34** (1) (1928), 115-116.

In view of the large number of texts on higher algebra already on the market, the average mathematician is not ant to hail a new book on this subject with joy or even interest. As one's eye runs hastily down the table of contents and sees a sequence of titles that calls to mind a composite picture of Weber, Serret, Burnside and Panton, et al., one can certainly be excused if one does not at first wax enthusiastic. ... first impressions lead one to expect merely one more book to add to the library catalogue and to which a beginning graduate student will look for help when Weber, Serret et al. happen to be in use. But, fortunately, first impressions are unjust in this case. For there are two places in the book where the treatment is sufficiently different from the classic treatment to interest a reader familiar with the usual treatise on the subject. Probably the outstanding characteristic of the book is the prominence which is given to the notion of field (domain of rationality). This is certainly good pedagogy, for this is one of the most important concepts in mathematics. ... Also, the emphasis on fields in the first volume admirably prepares the way for the discussion of Galois' theory of equations in the second volume. Although the notion of field is always used in this subject, of necessity, yet the author's presentation is such as to give to the student a number of important theorems about algebraic fields which he does not usually see formulated outside a course in algebraic numbers and should give him a grasp of the elements of this subject such as he does not usually have after taking the usual course in Galois' theory of equations. This aspect of the book is highly to be commended.

**10. Algebra Second edition (volume 1, 1932, volume 2 1933) by Oskar Perron.**

**10.1. Review by: Mina S Rees.**

*Amer. Math. Monthly*

**40**(8) (1933), 484-485.

The first edition of this two-volume work, which has for several years had a recognised place in our libraries, was distinguished by the prominence given to the field concept. In the new edition there is an improved presentation of this concept and its fundamental significance in many parts of the theory is further emphasised. The author retains as his objective the desire to write a book available to a beginner, but one which takes account of recent research in the subject. ... Insistence on clarity and correctness in the presentation of details make this a good text for beginning students, though the more experienced reader may feel at times that the author has been overscrupulous in the inclusion of all details. Professor Perron has succeeded admirably in preserving the high standard of the first edition, while making revisions which should enable the student to read with greater ease current algebraic research.

**11. Nichteuklidische Elementargeometrie der Ebene (1962), by Oskar Perron.**

**11.1. Review by: Hans Schwerdtfeger.**

*Mathematical Reviews*MR0139049

**(25 #2489).**

This book gives an introduction into plane non-euclidean geometry for the benefit of teachers and students with a sound knowledge of high school geometry. ... The book is the outcome of lectures and earlier work of the author. In the derivation of the geometry itself the models are not used in order to avoid having non-euclidean geometry appear as a special chapter of euclidean geometry; to repeat Coxeter's statement: it is the "purest" treatment of non-euclidean geometry. Its more formal sections demand solid concentration; but there are rest periods conveying plenty of helpful information in a pleasing manner.