## Reviews of Kenneth May's books

Kenneth May wrote two textbooks aimed at undergraduates. We give below brief extracts from six reviews of these innovative books.

**Elementary analysis (1952), by Kenneth O May.**

**1.1. Review by: R Walker.**

*The Mathematical Gazette***37**(322) (1953), 312.

This volume, which was first published in 1950 under the title Analysis, a Freshman Course, is designed to provide "a unified treatment of material usually labelled algebra, analytic geometry, trigonometry, and introductory calculus. The book is directed to those who are going to use mathematics in advanced courses or in science, engineering, or business. Although it assumes as a minimum only one year of high school algebra and one of plane geometry, there is plenty of material for a year's work by students with more extensive preparation".... The treatment throughout is very detailed and elementary, and particularly so in the first half of the book. ... But though elementary and discursive, the book is thoroughly sound and could profitably be read by anyone intending to continue the study of mathematics at a more advanced level. Unfortunately, for so elementary a book, its price in this country is high. There are over six thousand exercises, worked and unworked, many of them of practical interest. Answers and tables are provided.

**1.2. Review by: W N Huff.**

*The American Mathematical Monthly***59**(10) (1952), 708-709.

This book is a unified treatment of algebra, trigonometry, analytic geometry and some of the concepts of calculus. It is by no means just another book in this field but is a thoroughly wrought piece of text-book writing. At least one year of high school algebra and plane geometry are supposed. The book is adaptable to students wishing a thorough grounding for calculus and beyond and to students wishing a terminal course. This last class of students can get some of the important ideas of mathematics which have given the subject its use and beauty. The author gives the teacher useful suggestions in making up a course. ... It is clear that the author has spent both time and care in getting together a great many problems chosen to gain manipulative practice and illustrate the text material. The book is a vast store house of problems well chosen from the uses of mathematics in the physical sciences, economics and business. A teacher using the book should be on the lookout for the many exercises extending the theory. Included among such are important loci on polar coordinates and problems on various properties of the conics. ... The author's style is lively and clear and his exposition excellent. ... Each college and university offering a unified Freshman course has its own particular problems. The reviewer believes that careful consideration of various courses designed from this well-written book may help to find a solution of such problems and to provide a first course in college mathematics which is beneficial to students and enjoyable to teachers

**1.3. Review by: F W Kokomoor.**

*The Mathematics Teacher***46**(8) (1953), 608.

Students who master the content of this book will have a very satisfactory equipment in first-year college mathematics. The book is packed - almost overstuffed - with good material. It is intended for students with a minimum preparation of one year each of high school algebra and plane geometry, but the author says that there is ample material for students with more extensive preparation, and this is by no means an overstatement. ... On the whole, this is a very creditable book, done by one who knows how to organize his work. To those who are looking for a solid treatment of first-year college mathematics in unified form, we recommend a consideration of this book.

**1.4. Review by: R L Anderson.**

*Econometrica***21**(1) (1953), 214-215.

Too many of our elementary mathematics texts have been written for the engineer or the physical scientist with emphasis on practical problems rather than on an understanding of mathematics as a logical structure. Professor May has written a book which teaches mathematics for its own sake. He first shows how the number system was developed. Then the various functional relationships are considered, along with the necessary mathematical tools; this is a welcome departure from the usual procedure of discussing the tools and then illustrating their usefulness with various functions. The functions are considered in this order: linear, quadratic, power, exponential and logarithmic, and trigonometric. The author concludes with a number of topics which were too difficult or specialized to include earlier: analytic geometry (including parametric methods, vectors and polars), complex numbers, conic sections, polynomials (roots and the calculus), curve plotting, and functions of two variables. Two special topics worthy of mention are included: inequalities and method of induction.

**Elements of modern mathematics (1959), by Kenneth O May.**

**2.1. Review by: Robert C Seber.**

*The Mathematics Teacher***53**(1) (1960), 39-40.

This introductory textbook for college students cannot be used casually. If it is read care fully, one will find that its author has made an extraordinary effort "to help the student obtain a modest mathematical literacy." In doing this the text represents a trend to emphasize concepts and understandings while utilizing modern mathematics. The high school or college instructor interested in exposition which reflects established content in the light of recent mathematical literature and applications of mathematics will find many things of interest in this book. To actively engage the student with the ex position of the text, lists of questions, called exercises, are included within the text at many key points. These questions deal with ideas which are essential to the development of the text and which are frequently sources of difficulty for students. When a student comes to class or to the instructor for help with a not too uncommon "I don't understand," and the searching for what he does not understand begins, the exercises in this text can provide excellent help. To assure that ideas initially grasped through the exposition and exercises are brought into use, extensive lists of problems are provided at the end of each section. These problems run from those of a routine nature to the type which could occupy a student for several hours, or weeks, or which, in some cases, could lead him into an area which might challenge him professionally. In addition to the problems which are related specifically to concepts of mathematics, there are many problems with references to their sources in the biological, physical, and social sciences. By selection from the problem lists the instructor should be able to establish the type of student proficiency usually used to establish that learning has occurred. ... This book has many ideas to offer a student beginning his study of college mathematics. It will make demands of his talents and energy, he will probably not find it easy to read, but he should find that his efforts will be rewarded by an awareness of the nature of mathematics and some of its fundamental concepts upon which he can build a continued study of the subject.

**2.2. Review by: W R Scott.**

*The American Mathematical Monthly***67**(3) (1960), 309-310.

This first-rank book represents a major departure from present freshman level texts, even those of the "modern" school. The book first introduces some of the ideas and notation of logic and sets, and then proceeds to use them for the remainder of its 600 pages. One finds no mere lip service to these topics here; they are put to daily use by the student throughout the course. Other material treated includes plane analytic geometry, a start on differential and integral calculus, probability and statistical inference, and an introduction to abstract algebra. The strong point of this carefully and well-written book is the author's insistence that things be done right. ... The reviewer feels that Elements of Modern Mathematics is an excellent book for the good student and the good instructor, but that it may be somewhat ambitious for the average. Certainly any student who masters a course taught from this book will gain far more mathematical maturity than he would from any other text on this level with which the reviewer is familiar.