Algebra of Analysis (1944), by Karl Menger. **1.1. Review by: Cecil James Nesbitt.**

*Amer. Math. Monthly***52**(5) (1945), 271.The author takes a new approach to the theory of functions of a single variable by starting with a system of elements (to be called functions) which form a tri-operational algebra. The operations of this algebra of functions are addition, multiplication and substitution. The commutative, associative and distributive laws for these operations which are satisfied by ordinary functions are postulated. The author defines neutral or identity elements in regard to each operation, the neutral element for the substitution operation corresponding to the function y = x in the ordinary function theory. Topics covered include the theory of constant functions, the theory of inverses (where these exist) for the three operations of the algebra of functions, definitions and properties of exponential, logarithmic, power and trigonometric functions, the algebra of calculus including anti-derivatives. The last chapter explores the concept of algebra of functions of several variables. The booklet is written in a fluent style and it can be readily appreciated by a reader who has an elementary knowledge of the concepts of ring and field. The limited distributivity available in regard to the substitution operation impedes the development of any very deep algebraic theory for the algebra of functions. In some topics, the author's notation and method achieves considerable elegance; this is, however, balanced on occasion by formalism and expedient postulation.

**1.2. Review by: Alonzo Church.**

*The Journal of Symbolic Logic***10**(3) (1945), 103.Special relevance of this to the field of this Journal is tenuous. It consists in some resemblance in the author's method to that of combinatory logic, and in the reviewer's tentative conjecture that the treatment might be improved by more extensive adoption of some of the ideas of combinatory logic. The treatment of functions of several variables and partial derivatives by the author's method is barely begun in the last six pages of the mono-graph, yet the array of kinds of operations of substitution upon functions (as compared to the one such operation, fg, required in the case of functions of one variable) is already formidable. An analysis of these operations into simpler constituents seems to be called for. Moreover the author's device of taking constants to be functions of a particular kind may lose some of its effectiveness through his failure to take the next step of treating singulary functions as binary functions of a particular kind, binary functions as special cases of ternary functions, and so on (so that all elements of the system are of the same rank or category).

**1.3. Review by: Orrin Frink.**

Mathematical Reviews, MR0011280**(6,142g)**.This lithoprinted booklet gives an abstract treatment of the part of analysis which involves limit concepts only superficially and can therefore be treated algebraically. The first chapter describes an abstract algebra of functions of one variable (although variables do not occur explicitly). ... The second chapter, on the algebra of derivatives and integrals, studies the differentiation operator D and its right inverse S abstractly. ... The third and last chapter on functions of higher rank lays the ground-work for a theory of partial derivatives and of functions of several variables. The booklet is written in an informal style which makes for easy reading.

Calculus: A Modern Approach (1952), by Karl Menger. **2.1. Extract from the Preface.**In this book we define a variable quantity in terms of the concepts of class and number. We develop a strictly deductive theory of variable quantities. We describe in detail the application of the results to physical science, where variable quantities are of paramount importance. We insist on a clear-cut distinction between a variable quantity and a Weierstrass variable. We attribute some of the difficulties which Russell left unsolved, and some of the confusion in the mathematical literature, to a lack of such a distinction. We deny that the theory of variable quantities as used in science was arithmetized by Weierstrass, and claim that it cannot be arithmetized. We outline a postulational treatment of the theory of pure functions. Finally, we bring out an arithmetico-analytical parallelism between the mensuration of objects and the functional relation of variables - unit and independent variables being analogues.

**2.2. Review by: Merrill Edward Shanks.**

*The Scientific Monthly***77**(1) (1953), 56.This stimulating book is intended as a textbook in a beginning calculus course, a point of view which will arouse numerous cries of protest as well as a much smaller number of assents. It is the author's thesis that the conventional notation of calculus tends to obscure the essential ideas and make the subject excessively difficult for the beginning student. Professor Menger has systematically developed a notation for functions and their derivatives and integrals that he claims to be superior not only logically but also pedagogically. Whether or not his notational innovations will be accepted in their entirety seems to be doubtful, but certainly many of them are highly worthy of consideration by all teachers and writers. The book is recommended reading for teachers because of its unorthodox point of view and the freshness of its presentation. That the book is convenient to use as a textbook in an elementary class seems questionable but the reviewer is sympathetic with the author's aims and approach and would like very much to try it. ... Professor Menger is to be congratulated for producing a stimulating book which should have a permanent effect on the teaching of elementary mathematics.

Calculus: A Modern Approach (2nd Edition) (1953), by Karl Menger. **3.1. Review by: Gerald James Whitrow.**

*The British Journal for the Philosophy of Science***9**(34) (1958), 172-173.Although the principal object of this book is to bring about a revolution in the teaching of the calculus, its interest and significance is philosophical as well as pedagogic because of the author's discussion of the ideas of function and limit. The author defines the word ' function' and not the usual phrase 'y is a function of x' which raises the question of the nature of x and y. He defines a function as a class of pairs of real numbers in which two pairs which have the same first element have also the same second element. Apart from certain differences in notation from the traditional method, notably dropping brackets and writing fx for f(x), this definition makes it possible to distinguish clearly between the function f itself, which is a class of pairs of numbers, and the value of the function fx at x, which is a number. ... There is much else in the book which challenges the customary notation and exposition of the calculus, both differential and integral, but discussion of these features hardly falls within the purview of this Journal. There is no doubt that this is a book of the greatest pedagogic importance. Owing to the marked divergence between the author's methods and those which are now current, it is unlikely that his will be generally followed, at least in the immediate future. Nevertheless, this is a book which should be read by all who teach the subject for it will give them fresh ideas and inspiration.

**3.2. Review by: Steven Orey.**

*The Journal of Symbolic Logic***24**(3) (1959), 222-223.This is a new edition. It is an unusual calculus text in many respects. ... The word 'function' is reserved for many-one correspondences between two classes of real numbers. A many-one correspondence between an arbitrary class and a class of reals is called a 'fluent'. It is explained that the proper analysis of applications of mathematics usually involves the introduction of certain fluents defined on a space. ... The book strongly insists on certain innovations in notation for the calculus. Some of these, e.g. the systematic use of a symbol for the identity function, do indeed seem very convenient.

**3.3. Review by: Orrin Frink.**

Mathematical Reviews, MR0066439**(16,575g).**This is a calculus textbook for undergraduates incorporating innovations in notation and terminology which the author considers necessary and sufficient for the avoidance of ambiguities and errors associated with more conventional notations. ... By a "variable quantity" the author means a real-valued function defined over an arbitrary set, and by a "function" he means a real-valued function defined over a set of real numbers. Since this is a text-book for undergraduates, he does not give a completely rigorous presentation of his system. Aside from difficulties related to division by variable quantities which occasionally assume the value zero, he makes out a good case for the sufficiency of his system. ... He makes out ... a less good case for its necessity.

Calculus: A Modern Approach (3rd Edition) (1955), by Karl Menger. **4.1. Review by: Reuben Louis Goodstein.**_

*The Mathematical Gazette***41**(335) (1957), 79.This book presents a development of the ideas introduced in 1944 by the author in his Algebra of Analysis. An earlier edition of Menger's Calculus was reviewed by Dr Busbridge. Menger's treatment is characterised by a complete revision of the traditional uses of variables, a revision which he has recently described in his article in the Gazette called "What is x and y". That there are serious confusions in the traditional uses of variables is beyond doubt, and much credit is due to Dr Menger for his criticism; but it may well be asked whether an elementary text book on the Calculus is the best place in which to feature these criticisms and the notational innovations to which they have led the author.

Géométrie Générale (1954), by Karl Menger. **5.1. Review by: Thomas James Willmore.**

*The Mathematical Gazette***39**(327) (1955), 71.This book contains a general survey of advances made during the past twenty-five years in the subject of generalised metric geometry. A very considerable amount of information has been packed into these eighty pages, and the book forms a very valuable addition to the literature of metric spaces. The treatment is concise but lucid; and there is an extensive bibliography of about a hundred items where the reader is referred to original works for de-tailed proofs of results and theorems stated in the text. Since the introduction of the notions of filter and uniform structure, it has been recognised that metric spaces do not play such a vital role in the development of general topology. However the view taken in this book is that the metrical properties of general metric spaces are interesting in themselves.

**5.2. Review by: Leonard Mascot Blumenthal.**

Mathematical Reviews, MR0058229**(15,340d)**.The brief review of selected topics of Distance Geometry contained in this Mémorial formed the basis of lectures given by the author at the Sorbonne in 1951. It is similar in style and spirit to earlier short summaries of distance-theoretic results published by its author.

The Basic Concepts of Mathematics (1957), by Karl Menger. **6.1. Extract from Preface.**Algebra, analytic geometry, and large parts of the calculus, as taught today, are products of the 17th century. The fundamental ideas of those branches of mathematics are among the great legacies of that period and have ever since belonged to the most precious heritage of mankind. But with those ideas, the 20th century has also inherited the form in which they were presented by their discoverers. It has inherited the noun variable in several meanings that are completely at variance with each other, the noun constant in uses that are blatantly self-contradictory; and terms such as parameter and indeterminate that are utterly obscure. It has further inherited the indiscriminate use of the letter

*x*in more than ten altogether discrepant types of procedure-Renaissance mathematicians practised some kind of*x*-olatry. Because all those terms and the symbol*x*have not been, and are not being, used consistently, their vocabulary and grammar have never been written. This is why those crucial chapters of the theory of the mathematical language have not been, and are not being, taught and why, in the process of learning mathematics, important meanings and rules must be surmised. The deepest difficulties of mathematical education thus are not due to shortcomings in education. They are due to procedures in mathematics.**6.2. Review by: Julius Richard Buchi.**

*Philosophy of Science***24**(4) (1957), 366.There are two aspects from which the basic concepts of mathematics can be viewed: (1) logical foundations, (2) teachability. In this booklet Menger is mainly concerned with the teaching of mathematics. He rightly holds that the many inconsistencies in terminology and symbolism, which are carry overs from the creative period of the seventeenth century, are among the main obstacles to students of college mathematics. In this booklet (a) he points out the most glaring of these inconsistencies, and (b) he suggests a remedy for them. In the reviewer's opinion, Menger has well succeeded as far as (a) is concerned, i.e., he has located the worst trouble spots in the conventional teaching of mathematics. There-fore, a careful study of this booklet is recommended to both students and teachers of college mathematics. However, as far as (b) is concerned, in the reviewer's opinion it is unfortunate that Menger does not seem to realize that aspects (1) and (2) are not entirely independent. It is not clear whether Menger's ad hoc suggestions for a new way of teaching mathematics can really be carried through systematically.

**6.3. Review by: Earl Hicks Crisler.**

*Amer. Math. Monthly***64**(8) (1957), 603-604.In the words of the author, "this book .. . is addressed to everyone desirous to clarify his ideas about some of the basic concepts and fundamental procedures of mathematics." The chapter headings indicate the procedures and concepts treated. They are: Numbers and Numerals, Facts and Formulas, Variables in General Statements, Unknowns in Equations, Variables in Description of Classes, Parameters in General Problems, General Expressions, Complex Numbers, Indeterminates in Polynomials. A particular goal of the book is to resolve the many difficulties that arise through the use of the words "variable", "indeterminate", "parameter", "constant". Professor Menger accomplishes this aim with charm and skill by means of numerous examples and also by references to the words' mathematical origin and subsequent history. The book's tone is conversational and informal. Examples are discussed in detail and exercises are provided at appropriate places. It is a very readable introduction to mathematical semantics. The author has accomplished his objective in that this book should appeal greatly to the persons to whom it is addressed.

**6.4. Review by: Joseph Henry Woodger.**

*The British Journal for the Philosophy of Science***9**(34) (1958), 172.This book can be warmly recommended to all who teach or learn elementary mathematics. ... [It] is primarily addressed to beginners, it is anticipated that it will also interest the accomplished mathematician and scientist; they will at least be interested in the way the author overcomes the difficulties mentioned above. He himself appeals 'to everyone desirous to clarify his ideas about some of the basic concepts and fundamental procedures of mathematics'.

**6.5. Review by: Herbert Edward Vaughan.**

*The Mathematics Teacher***51**(3) (1958), 208.This pamphlet should be studied by every teacher of mathematics from those who introduce first graders to the beginnings of arithmetic, through those who direct graduate studies. For some time, Professor Menger has stressed the need for greater precision in the language used in mathematics, and he now at tempts to show how this need can be satisfied insofar as the language of arithmetic and algebra is concerned. His attempt is sufficiently successful to justify, in the reviewer's mind, the emphatic recommendation which heads this review. However, after making such a recommendation, the reviewer would be remiss if he did not discuss at least some of the relatively few points in which he disagrees with the author. ...

Studies in Geometry (1970), by Leonard M Blumenthal and Karl Menger **7.1. Review by: Hans Freudenthal.**

*Amer. Math. Monthly***78**(3) (1971), 315.This is a very elementary, meticulously formal introduction into a few chapters of geometry which in its kind can hardly be surpassed, with many useful exercises to develop by little steps reasoning in these fields. One of the authors in his preface recalls the famous words that Plato wrote on the entrance gate to his academy: "Let no one unacquainted with geometry enter here." The referee would prefer here to quote another porch inscription "Lasciate ogni speranza voi ch'entrate," though there is some hope left for the reader that after a longwinded path through 'Inferno" and 'Purgatorio' he enjoys the 'Paradiso' of geometry.

**7.2. Review by: Heinrich Walter Guggenheimer.**

Mathematical Reviews, MR0273492**(42 #8370).**The book is very aptly described by its title. It is an introduction (Parts 1 and 3 by the first author, Parts 2 and 4 by the second) to the work of the authors on geometry based on lattices and metric spaces. In these fields it contains a wealth of material in easily accessible form that is not otherwise available in book form. On the other hand, closely related modern developments are not mentioned.