Basic Mathematics for War and Industry (1944), by P H Daus, J M Gleason and W M Whyburn. **1.1. From the Preface.**The war program has given rise to extensive training and educational activities designed to meet needs of essential industries and the armed forces. Skill in the use of mathematics, particularly arithmetic, algebra, geometry, and trigonometry, is an integral part of all phases of a highly technical war. Similarly, but to a less spectacular degree, basic mathematical skills are indispensible to peace-time activities. The experiences of serious investigators lead to the uniform conclusion that the same principles of elementary mathematics are needed for the armed forces, for war industry, and for ordinary civilian activities. ... This book is written to provide a single text in which selected principles of elementary mathematics are presented in a carefully organised manner. The book has been made thoroughly practical - both in the choice and treatment of topics. The practical aspects have been introduced without sacrifice of mathematical rigour or accuracy of statement.

**1.2. Review by: Herbert A Simmons.**

*National Mathematics Magazine***18**(6) (1944), 253-254.This book presents the basic facts of arithmetic in 30 pages, of algebra in 53 pages, of plane geometry in 67 pages, of plane trigonometry in 51 pages, and of solid geometry and spherical trigonometry in 45 pages. ... The exposition of the text is clear. One would not ask for better explanation of the principles covered. The exposition is occasionally novel. ... One of the best things about the book is its problem lists. The problems contain an enormous amount of information from physics, mechanics, engineering, and other subjects. We believe that most any college teacher will find refreshing problems in this text.

**1.3. Review by: Lousie A Wolf.**

*Amer. Math. Monthly***51**(8) (1944), 469.This text provides in a single volume topics of elementary mathematics which are usually found in separate text-books. ... The exposition is surprisingly detailed when one considers how much material is covered in the book. The typography is clear and the numerous figures are excellent. As might be expected in a condensed text, many proofs are omitted and some propositions from geometry are called axioms. However, some of the brevity is attained by presenting clever short proofs and by avoiding the formal superposition and statement-reason types of proof. The illustrative problems and the exercises throughout the book draw upon all fields of applications with special emphasis on aerial and naval navigation, physics, and geometry. An exceptional feature is the use of spherical trigonometry and spherical geometry in the solution of shop problems.

First year college mathematics with applications (1948), by P H Daus and W M Whyburn. **2.1. From the Preface.**This book is written with the intention of providing a single text for the study of first year college mathematics, especially in engineering and technical schools. Its purpose is to provide a strong and ample background for the study of the calculus, and to integrate the subjects of college algebra, analytic geometry, and analytic trigonometry. At the same time the text illustrates all principles by applications taken from science and engineering, so that the course is completely independent of its future use.... Suggested lesson outlines for a ninety-hour course (three days a week for two semesters) and for a seventy-five-hour course are given. ...

**2.2. Review by: Michael E Beesley.**

*Amer. Math. Monthly***57**(2) (1950), 124-126.Those who prefer to give relatively strong training in analytic geometry should welcome this book with enthusiasm. In sharp contrast with recent works which combine analytics with calculus and thereby effect a de-emphasis of the former subject, this text retains most of the traditional geometrical topics. Moreover, it provides a vigorous and well-motivated presentation which, in the opinion of the reviewer, is superior to that of many established books.

**2.3. Review by: Ezra J Mishan.**

*Economica, New Series***27**(108) (1960), 375-376.Daus & Whyburn's book [is] little more than an introduction to the calculus, but more terse and formal in treatment than is perhaps suitable for a beginner. As the authors say in the preface, it presupposes a second course in algebra, preferably taken in college. Though the treatment is not difficult, it is likely to be more readily intelligible to those with some mathematical training who are seeking knowledge of economics than for economists without any initial facility in manipulating simple mathematics.

**2.4. Review by: Marion E Stark.**

*Mathematics Magazine***23**(5) (1950), 268-269.This book is planned particularly for engineering and technical schools. Calculus and determinants are omitted. ... The authors are clear and accurate on one-to-one correspondence, on curve tracing of all sorts (in spite of the absence of the derivative), on systems of straight lines and of conics, in the introduction of "sin

^{-1}x", in the discussion of parametric equations and the reasons for them, in the use of hyperbolic functions, in the introduction of occasional bits of history (i.e. the "three famous problems of antiquity"), and so on. Examples are good and plentiful, especially those involving Physics. The reviewer is cheered to find that most of analytic geometry as usually taught in a college of liberal arts is satisfactory for engineering and technical schools. ... A few sentences are so good that they must be quoted. "It is more important to understand the complete analysis of any of these problems than to carry out all numerical calculations for some of them" (p. 151). "But the method is of more importance than the formula" (p. 170). "Any attempt at point-plotting requires the solutions of cubic equations with irrational roots, and this is discouraging, to say the least" (p. 278). "For purposes of prediction the use of an assumed law, without some evidence other than statistical data, is fraught with dangers and has many times led investigators to erroneous conclusions" (p. 314). Why not write that last sentence in letters of gold upon the walls where all would-be statisticians may read them!Algebra for College Students (1955), by W M Whyburn and P H Daus. **3.1. From the Preface.**The need for a college course designed to strengthen the background of a high percentage of entering freshmen before they understand university mathematics courses is acknowledged by many departments. The authors of the present book have observed that difficulties experienced by college mathematics students result not only from weakness in the manipulative skills of algebra, but also from deficiencies in geometry and arithmetic and the lack of correct habits of thinking and reasoning. They have further observed that a college course in intermediate algebra often consists of an unenthusiastic rehash of material that the student has already surveyed at least once, and often several times, before. Absence of fresh mathematical ideas makes for low student and teacher morale. Under such circumstances, results of the course are likely to fall short of maximum desires and expectations. This book endeavors to include those portions of arithmetic, algebra, and geometry in which the student's background may be weak. At all points it tries to present the subject matter in a way that will stimulate the alertness of both student and teacher.

**3.2. Review by: Irwin K Feinstein.**

*The Mathematics Teacher***48**(8) (1955), 571.The authors have tried to write an intermediate algebra for college students, which would differ from an adult approach to college algebra only in the range of the topics covered. For this reason topics in the areas of theory of equations, permutations, combinations and probability, complex numbers, partial fractions, topics usually treated in college algebra text books, are not found in this text. Included, how ever, are some excellent materials not usually found in intermediate algebra books - the basic ratios and identities of trigonometry, the sine law, the cosine law, mathematical induction, series, a brief discussion of the number e, short introductions to algebra of matrices, and Diophantine equations, among others - materials which the inspired teacher will welcome. The exercises throughout the text, although some what few in number, are excellent and varied, each one requiring the student to extend his previous learnings. ... his text should be welcomed by most teachers and disliked by few. It is time for intermediate algebra to become one of the firm bases on which mathematics builds. This can never happen if textbooks continue to hash and rehash the manipulative and computational aspects of material which the college student has had in his earlier work, often times more than once. The fresh mathematical ideas and view points, the excellence of the problems, the clear ness with which the basic ideas are presented, the maturity with which the basic material is treated - all of these should make for high teacher and student morale in the area of inter mediate algebra.

**3.3. Review by: V D Gokhale.**

*Amer. Math. Monthly***63**(4) (1956), 265-266.As the authors remark in the preface, the "difficulties experienced by college mathematics students result not only from weaknesses in the manipulative skills of algebra, but also from deficiencies in geometry and arithmetic and the lack of correct habits of thinking and reasoning." These incorrect habits are often inherited by the students from the teachers in high school mathematics. The reviewer recalls a remark of a high school teacher in one of his classes, "I do not see what logic has to do with mathematics." Texts like the Chicago syllabus or 'Principles of Mathematics' by Allendoerfer and Oakley (McGraw-Hill) are specially aimed at this deficiency. The present text is a compromise between such texts and the usual college algebra texts which are often "an unenthusiastic rehash" of high school mathematics.

Introduction to Mathematical Analysis with Applications to Problems of Economics (1958), by P H Daus and W M Whyburn. **4.1. Review by: Robert Wayne Clower.**

*Econometrica***27**(4) (1959), 715-716.This book is apparently intended to fill the need - if one exists - for a modernized, abbreviated, and Americanized version of Allen's classic 'Mathematical Analysis for Economists'. It starts later, stops sooner, and aims lower than Allen's work (except that it includes a chapter on curve fitting); but it is more modern, more concise, and distinctly less English. As far as it goes, moreover, the argument is accurate and clear, balanced in content, technically solid, and remarkably free of typographical errors. ... the exercises are excellently conceived, numerous, and accompanied by answers. Of course the book does not go very far; in particular, it does not deal with such topics as matrix algebra and linear programming. Except for some perspicacious (but scattered) critical references to the existing literature on mathematical economics and econometrics, moreover, the analysis is not as philosophical in tone as might be desired in an introductory work. Hardly a word is said about the nature of mathematical reasoning or about the concrete interpretation of mathematical systems; nor is anything more than a sketchy rationale provided for the development and use of particular mathematical techniques. For these reasons the work is not particularly suitable for self-study purposes. In fairness to the authors, however, it must be said that the book is recommended for use only when courses in economics are taken simultaneously and, by implication, only when the curriculum is well planned and staffed by thoroughly competent people. The trouble is that such situations are extremely rare at the undergraduate level for which the book is specifically designed. As a practical matter, therefore, I suspect that the text will be useful mainly as an exercise book for graduate students and instructors whose grasp of the formal logic of economic analysis is something less than certain. In this capacity, however, the book should fill a definite need, and fill it well.

**4.2. Review by: Gerhard Kade.**

*Journal of Economics and Statistics***172**(5) (1960), 466-467.The increasing mathematization of economic theory always leads more to the need of additional mathematics teaching for economics students. Often already in the early semesters such students have noticeable difficulties in applying knowledge of advanced mathematics learnt at school to the analysis of economic problems. This book is composed by two mathematicians in a style of teaching the material specifically designed to meet these transitional difficulties. ... Within this limited framework, which is aimed especially at the initial semester, the authors offer a didactic performance giving many breakthroughs in understanding in the combination of mathematical methods and economic problems. Particularly noteworthy is the vivid style and the abundance of exercises (with solutions) that make it especially suitable as a self-study book.

**4.3. Review by: PierCarlo Nicola.**

*Rivista Internazionale di Scienze Sociali*(III)**40**(77) (5/6) (1969), 586-587.The volume in question seems particularly interesting for Italian students of Economics; in fact, it constitutes a clear and elementary introduction to traditional mathematics, the introduction is presented in a very simple way but not without rigour, and therefore fills a gap in the range of books, because every argument put forward is always illustrated by examples with elementary mathematics drawn from economics. Indeed, the examples treated cover a sufficient range to suggest that preparatory elements of economic theory are sufficiently illustrated here.

**4.4. Review by: Cletus Oakley.**

*Amer. Math. Monthly***67**(2) (1960), 196.This book covers such material as elementary notions of analytic geometry, differentiation and integration, partial differentiation, maxima and minima including constraint conditions and the method of Lagrange multipliers, curve fitting ... and correlation. Spread throughout these mathematical topics are the applications to economics, such as supply and demand curves, market equilibrium, effect of taxation, elasticity, marginal revenue, profit under monopoly, production functions, indifference maps, utility index, relative performance etc. The material is well written and the large number of interesting problems are more than adequate for a one-semester course. Especially recommended for students of business and economics.

Algebra with Applications to Business and Economics (1961), by P H Daus and W M Whyburn. **5.1. Review by: Gerhard Kade.**

*Journal of Economics and Statistics***175**(3) (1963), 278-279.Introductions to mathematical methods of economic theory have recently been more numerous - a clear sign of the conviction that the educational problems of mathematical teaching of economics and business administration are not considered as being solved. The mathematical models of economics have also been particularly well developed in the last two decades ... Daus and Whyburn give an introduction to the algebra particularly required to understand the basics of the methods of analysis described in the book by the same authors (Introduction to Mathematical Analysis with Applications to Problems of Economies, 1958). The largest part of this book deals in great detail with the methods of elementary algebra; the survey is therefore suitable mainly for repetition, in which the reference to economic applications increases the clarity of presentation. Particularly useful and understandable to the novice is then the constructible introduction to matrix algebra and the basics of linear programming. Corresponding to the entire system of this book, this introduction is very clearly illustrated by a variety of examples; Exercises (with solutions in the Appendix) make this textbook also suitable for self-study.