## Reviews of Cyrus Colton MacDuffee's books

**The Theory of Matrices (1933), by C C MacDuffee.****1.1. Review by: Mark H Ingraham.**

*Bull. Amer. Math. Soc.***40**(5) (1934), 372-373.This work definitely is of the encyclopaedic type though, owing to the author's search for elegant proofs and to the necessity of making each theorem depend on preceding work, there has been brought about a considerable amount of unity. Though this book is not lacking in original material, the author's personal contributions play a minor part and have been mostly published elsewhere. ... No mathematical library can afford to be without this book. Every worker in matrix theory will find it both enriching to his knowledge and a great timesaver. With all this the author has had the saving grace in each portion to make the reader feel how much is left to be done both by way of solving extremely difficult problems and by way of rather routine and methodical progress. In fact, the reviewer shudders to think how many doctors' dissertations will be the result of this definition of the boundary of knowledge in this one field.

**An Introduction to Abstract Algebra (1940), by C C MacDuffee.****2.1. Review by: Garrett Birkhoff.**

*Science, New Series***93**(2408) (1941), 185-186.A conspicuous mathematical development of the last two decades has been the growth of algebra as a unified science, fruitful in applications to modern physics and chemistry, as well as to other branches of mathematics. The general methods and results of the newer algebra were first made available to professional mathematicians as a whole by van der Waerden's now classic "Moderne Algebra." However, besides being written in a foreign language, this book was far too advanced and compendious to be a suitable text for a standard graduate course in this country. MacDuffee's new volume is the second noteworthy attempt to provide such a text, the first being Albert's "Modern Higher Algebra." Although it covers much less ground than Albert's book and contains no original material, MacDuffee's book, as its title suggests, affords an easier introduction to abstract algebra than Albert's. By emphasizing the most basic theorems and making no attempt at completeness, MacDuffee drives home the fundamental ideas of modern, algebra. And by illustrating each definition with carefully chosen examples, he gives "concrete" significance to them - a difficult feat in so-called "abstract" algebra. The book is not designed for purposes of reference or for use in advanced seminars.

**2.2. Review by: Lois W Griffiths.**

*National Mathematics Magazine***15**(4) (1941), 211-212.This book is intended to be used as a textbook by beginning graduate students. It will admirably serve this purpose, because it is very carefully written and is an introduction in fact as well as in name. The abstract point of view is developed gradually, as the author states in the preface. Concrete instances are presented before an abstract idea is developed

**2.3. Review by: Nathan Jacobson.**The purpose of this book is to serve as a text for a first graduate course in modern algebra. ... The book is very readable and the wealth of concrete examples should make it a useful text.

**2.4. Review by: Neal H McCoy.**

*Bull. Amer. Math. Soc.***47**(7) (1941), 539-543.The rapid advances in algebra within the last few years have been largely due to an exploitation of the powerful methods of abstract algebra. Accordingly there has been a tremendous increase in the interest in this subject, so that most colleges and universities offer at least one course in abstract algebra. However, students with only an undergraduate course in the theory of equations as a background in algebra frequently have considerable difficulty with the available texts - not so much in reading the proofs as in grasping the significance of the abstract theories. The present book was written primarily as a text for beginning graduate students and is designed to fill the gap between the usual text in the theory of equations and the more advanced texts on algebra. It should prove to be a valuable addition to the growing list of texts on abstract algebra.

**Vectors and matrices (1943), by C C MacDuffee.****3.1. Review by: Richard Brauer.**

*Bull. Amer. Math. Soc.***52**(5) (1946), 405-407.Professor MacDuffee's book is a clear and careful introduction to the theory of vector spaces and matrices. It should prove extremely useful not only to the student of mathematics, but also to the ever increasing circle of other scientists who show an interest in these fields. The more advanced undergraduate student will have no difficulty reading the book. The material is given in easy, almost leisurely steps. The book sets out from familiar facts, formulating the well known aspects in a new manner, gradually approaching new ideas, and almost inadvertently the reader will have become familiarized with the more abstract ways of mathematical thinking. There is, of course, another way of reaching the same aim: to push the reader into the cold waters of mathematical abstraction right away and let him swim around as well as he can. The student who follows Professor MacDuffee's book will not experience a sudden shock of this kind. Unless he ventures out into the last chapter prematurely (and in a footnote he is given permission to do so if he desires), he will learn about the more abstract concepts such as groups with operators, endomorphisms, and rings of endomorphisms only after he has digested the more "concrete" theories of vectors and matrices. ... Many students of mathematics will be extremely grateful to Professor MacDuffee for his well planned and lucid exposition.

**3.2. Review by: Neal H McCoy.**This is an exposition, from the vector point of view, of certain fundamental topics in the theory of matrices with elements in a field. The subject of central interest is the rational reduction of a matrix to canonical form, and the sequence of topics discussed leads naturally and easily to this goal. There are, however, a considerable number of by-products which are of interest in themselves.

**Theory of Equations (1954), by C C MacDuffee.****4.1. Review by: Leonard Carlitz.**

*Amer. Math. Monthly***61**(10) (1954), 723-724.As the author mentions in his preface, this book is the outgrowth of a one semester course given to junior and senior students. In a little over a hundred pages the author very competently covers the standard topics in the theory of equations. A small amount of abstract algebra is incorporated in the development of the subject. ... This is a pleasant and readable little book. The reviewer regrets that the author has not included more material even though there is ample material in the book for a one-semester course. Also a set of interesting miscellaneous problems would have helped round out the book.

**4.2. Review by: Cloeta G Fry.**

*Science, New Series***119**(3099) (1954), 730.This book was written for use as a text in a one-semester course in theory of equations. The mathematical maturity demanded of the student or reader is that of college juniors or seniors. The author intends the book both for those who plan to follow mathematics as a profession and those who will use it as a tool in other fields. He has used modern methods of algebra to pre- sent a complete study of linear systems of equations and of polynomials. The introduction of the abstract ideas of rings and fields forms a bridge between elementary algebra and advanced algebra and gives the student an insight into the more advanced work in modern algebra at an early stage in his training. These abstract ideas are introduced so naturally that they do not disturb the basic nature of the course, and a student who is not primarily interested in them will not object to them. ... At first glance, it may seem that there is not sufficient material for a one-semester course. But the abstract ideas of rings and fields cannot be pushed too fast in some classes, and an instructor may find that extra help and drill will need to be provided.

**4.3. Review by: Kurt A Hirsch.**

*The Mathematical Gazette***39**(329) (1955), 260.This little book has grown from a course the author has given repeatedly for students in their junior and senior, that is, third and fourth, year. It is ("therefore"?) quite suitable for our First Year Honours students. ... The treatment is confined to the most elementary aspects of the subject matter and avoids all deeper theorems. The title of the book is, therefore, a little misleading; for it does not do justice to the classical, nor to the modern theory of equations. But it can be recommended as a first introduction to university algebra.

**4.4. Review by: Kenneth W Wegner.**

*The Mathematics Teacher***47**(6) (1954), 422.This text is superior to many of its type in at least three respects:

(1) Topics which are usually covered in first-year college courses, such as systems of linear equations, are presented in such a way that the student will be aware of insights and understandings he missed as a freshman.

(2) The student will benefit in encountering the "arbitrarily small epsilon" notation of his calculus course in several places such as in proofs of Budan's and Sturm's theorems.

(3) Several modern algebra concepts, such as ring and automorphism, are woven in skilfully and naturally. The omission of determinants and matrices is a disadvantage in the instance where an additional course covering these topics is not taken.

JOC/EFR May 2013

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