1.1. Review by: Albert Nijenhuis.
Bull. Amer. Math. Soc. 68 (1962), 552-555

In the 1930's a rather interesting correspondence took place between the Dutch differential geometers J A Schouten and J Haantjes on one side, and their Polish colleague A Wundheiler on the other. The question at stake was what the most general (local) structures would be on a differentiable manifold, and how a theory of such things could be built up. A motivating factor in their discussion was the fact that - by the example of affine connections - tensors were not sufficiently general to cover all possibilities. The result, published in 1936-1937, was the definition of objects, macro-geometric objects and micro-geometric objects. All three have in common with tensors that finite ordered sets of numbers (or functions) - called components - are assigned to each admissible coordinate system. Objects were defined to be just that, while for the geometric objects the components with respect to two coordinate systems must be obtainable from each other if the relation between the coordinate systems is known. Roughly speaking, the macro-geometric objects remind us of sheaves, while the micro-geometric objects are reminiscent of fibre bundles. Most later literature, including the book under review, deals with micro-geometric objects only, and calls them geometric objects.

While attempts to formulate the theory of geometric objects in fibre terminology were made by Haantjes-Laman (1953), Kuiper-Yano (1955), and one with functorial overtones was made by the reviewer (1960), practically all of the 120 or so works on the subject (three-fourths of which by 8 authors) use the original terminology of Schouten-Haantjes and Wundheiler. To modern mathematicians this may seem cumbersome, but changing the terminology would not solve any hard problems, of course.

Really very little is known about geometric objects. The broad generalities are contained in the reviewer's thesis (1952), while the very special facts and results are the main concern of the book under review. The outstanding problem, according to the authors, is that of classification, to which essentially the whole book is devoted.
...

A number of results in the book are products of considerable ingenuity. Yet, anything even remotely resembling a complete classification is much too optimistic an aim. Geometric objects are just too easy to manufacture: take any eager student who knows the summation convention and other rites of tensor arithmetic and let him stir some tensor fields and partial differential operators; affine connections to be added according to taste. The result is invariably an object and chances are that by adjoining some intermediate steps of the random computation (for example, everything he has written down) one obtains a geometric object. On these grounds the reviewer believes that no classification attempts make much sense until, by suitable restrictions, the trivial has been separated from the significant. For example, it seems very unlikely that limiting the dimension $n$ of the underlying space to values such as 1 or 2 is really much help in this respect.

It is the reviewers personal opinion that geometric objects will never be of as central a significance as their inventors had hoped. On one hand, they serve as a suitable context in which to formulate general procedures (such as Lie differentiation); on the other hand, certain well-selected sub-classes will be worth classifying.

1.2. Review by: Albert Nijenhuis.
Mathematical Reviews MR0133763 (24 #A3588).

In the terminology of Ricci and followers, a tensor at a point $x$ of a manifold is a law which assigns to every admissible coordinate system a finite ordered set of numbers (called components), the components corresponding to any two coordinate systems being related by a linear representation of the Jacobian matrix at $x$ connecting the two coordinate systems. A geometric object is a more general entity, again with finite ordered sets of numbers (components) being assigned to admissible coordinate systems, but now the relation between different sets of components need not be linear, and the representation, while determined by the change of coordinates, may involve far more than just first-order derivatives. Historically, the name geometric object was meant to convey the wish that all local differential-geometric properties should be expressible in terms of objects no more general than these.

The material is presented in five chapters, and is a reasonably complete summary of the literature on the topics named in Chapters II through IV. The first, a general introduction, gives basic terminology and notations, with various examples, one of which is the well-known linear connection. ...  Chapter II is devoted to the involved problem of classification, which is only partially solved. ... Chapter III, on algebra of g.o., is in fact an investigation of geometric comitants. ...  Chapter IV, on covariant differentiation, deals with some aspects of differential comitants. Since the latter are hard to find for most g.o., the question is widened by admitting auxiliary g.o. to enter (classically for tensors: add a connection). ... The last chapter contains an enumeration of further problems, such as comitants of densities, scalar comitants of vectors, tensor comitants of vectors, Lie derivatives, etc., and some unsolved problems.

2.1. From the Foreword.

The solution of functional equations is one of the oldest topics of mathematical analysis. D'Alembert, Euler, Gauss, Cauchy, Abel, Weierstrass, Darboux, and Hilbert are among the great mathematicians who have been concerned with functional equations and methods of solving them. In this field of mathematics, as in others, the literature has grown markedly during the past fifty years. (See the chronological bibliography at the end of this volume.) However, results found in earlier decades have often been presented anew because through the years there has been no systematic presentation of this field, in spite of its age and its importance in application.

In this monograph, an attempt is made to remedy this situation, at least in part . Results are usually presented with proofs, in contrast to S Pincherle's German and French encyclopaedia articles published in 1906 and 1912, which, of course, were written for a different purpose. Earlier works (such as those by E Czuber 1891, E Picard 1928, G H Hardy, J E Littlewood, and G Pólya 1934, M Fréchet 1938, and B Hostinsky 1939) (see bibliography) also give some attention to functional equations, but the special functional equations treated are subordinate to their applications. We prefer to arrange the subject matter according to actual types of functional equations. We also cover a different and, as we think, somewhat broader range of problems than does the book of M Ghermanescu 1960. A R Schweitzer's plan of 1918 to compile a bibliography of the theory of functional equations was, alas, never carried out; therefore the list of references at the end of this book, although incomplete, can partly serve as a bibliography too.

The term functional equation is interpreted here in its modern, more restricted sense (cf. exact definition in the introduction, Sect. 0.1), so that the definition does not apply to differential, integral, integrodifferential, differential functional, and similar equations. As defined, however, the field of functional equations is still vast, and it was necessary to limit even further the material to be treated. Although our interpretation of this concept includes difference equations, we decided to omit them because many standard treatises on the subject are available. For consistency, and because ample systematic discussion is partly available elsewhere. those functional equations are also omitted in which all the unknown functions contain at least as many variables as the total number of independent variables in the equation; for example, all iterative equations have been left out. Since functional equations eliminated by this decision have entirely different methods of solution from all the others, this fact may be regrettable to some readers, but the omission was necessary to keep the size of the book within reasonable limits and to preserve systematic unity.

A number of factors were considered in organising the subject matter: Functional equations for functions of one or several variables, for one function, or several functions, simple and composite equations, different elementary methods of solution and reduction to differential and other equations, special applications of the equations, historical considerations, and the like. The classification is not rigid; some investigations may be considered as belonging in a particular chapter but are treated elsewhere because of their specific relationships. Functional equations for vectors and matrices of a finite number of dimensions are treated briefly as a link between equations with one and with several unknown functions of one and of several variables. On the other hand, equations for operators and functionals are not considered, since to do so would entail delving too deeply into functional analysis. As indicated by the title, there is no treatment of functional inequalities; an investigation of this would have to include, for example, the entire theory of convex functions. Within its framework, the book merely touches on the use of functional equations to define functions and their extension from the real to the complex region, their extension to matrices and other forms, their use in constructing functions of several variables by means of functions of fewer variables, and similar questions-c-all of them outside the scope of this book, which is limited mainly to methods of solution.

Certain restrictions are imposed in regard to the domains and the range of functions, as well as the "regularity" of the functions figuring in the equations; otherwise, for instance, most of the algebra would have to be included in the treatment of functional equations of associativity, transformation, and distributivity. However, algebraic structures with other laws are mentioned, and most of the publications dealing with them are included in the bibliography. Also, the independent fields of mathematics which are largely concerned with the solution and application of functional equations, such as the theory of continuous groups and the theory of geometrical objects, had to be omitted. In this connection, the booklet (J Aczél and S Golab 1960) on functional equations in the theory of geometrical objects might be mentioned. Nevertheless, we have included in the bibliography works in this theory which utilise principally functional equations. On the other hand, emphasis is placed on the relation of the discipline under discussion to algebra and to many "algebraised" fields of geometry (continuous groups, vector analysis, and the like). The broad fields of application, such as probability theory, non-Euclidean geometry, and mechanics, which have contributed greatly to developing the discipline of functional equations, should also play a significant role. In this book, however, the treatment of applications is subordinated to the equations, and preliminaries and consequences in these fields cannot be discussed in detail. Particulars of applications which are required for clarity but which themselves do not use functional equations are sometimes given in fine print, as are less important examples and more elaborate parts of certain proofs.

The book, aside from the foreword, introduction, concluding remarks, and bibliography, is divided into two parts. Smaller divisions include chapters, sections, and subsections. The chapters are numbered in sequence through the book, the sections are numbered within the chapters, and the subsections within the sections. Theorems and formulas are numbered with in the subsections. ...

Unfortunately, very few existence and uniqueness theorems are included, and very little is included about the influence of the form of the equation on the form of the solution, since such information is almost non-existent. When possible, we investigate the generality of given methods of solution. Hence, types of functional equations are investigated with more general methods of solution, in addition to special functional equations with individual methods of solution which have been mainly investigated up to now. As a result, a certain order and correlation are imposed upon this disorganised field, although the lack of a unified theory is still quite apparent.

2.2. Review by: Marek Kuczma.
Mathematical Reviews MR0124647 (23 #A1959).

This is the first systematic treatise about the theory of functional equations ever written. The purpose of the book has been to give an account of the present state of the theory, as well as to describe some general methods of solving functional equations. Such general methods were not known earlier (except, perhaps, for Abel's general method of the reduction of functional equations to differential equations) and are the main success of the author's investigations in this field. The book can also be very helpful as a survey of the theory of functional equations, although some results are not quite clearly formulated and a few facts that, in the reviewer's opinion, deserved to be mentioned, are not found there. The book is, however, supplied with an excellent bibliographical list, which contains over 1000 items and covers a period of over 200 years (1747-1960). But it is regrettable that the author makes no distinction between the more important papers and those less significant.

Of course, the author had to choose from a large subject, and in particular, to restrict suitably the notion of a functional equation. He assumes that a functional equation is an equality between two expressions formed from a finite number of superpositions of functions (at least one of which is unknown) and independent variables. Such a definition eliminates not only all differential, integral, functional-differential equations, etc., but also the functional equations occurring in the theory of dynamic programming. The further restriction is that the number of independent variables in the equation should be greater than the smallest number of variables on which depend the unknown functions. This restriction eliminates a wide class of functional equations, e.g., the difference equations, or those of Abel and Schroeder. The author has been justified, however, in omitting these equations from consideration, for there is a very deep difference between such equations and those dealt with in the book, and quite different methods are required in either case. Moreover, there is a great number of books about difference equations.

Also, the theories of continuous groups and of geometric objects, which are very closely connected with functional equations, are not treated in the book, since they form independent branches of mathematics. About the latter subject the author has written a special book, jointly with S Gołab, Funktionalgleichungen der Theorie der geometrischen Objekte (1960).
...
The whole book is written in quite an elementary style. Therefore, many interesting topics, which however require more advanced methods, are altogether omitted. But on the other hand, the book has the great advantage that it can be understood by anyone who knows the elements of the calculus.

Throughout the whole book, great stress is laid on the applications of functional equations. The applications in many diverse domains (Euclidean and non-Euclidean geometry and mechanics, mathematics of finance, theory of probability, characterisation of means, etc.) are described in detail. But in the reviewer's opinion, now, after this book has appeared, functional equations will be much more widely used, for only now the scientists have at their disposal a good treatise of the subject, which, as we hope, will be found very useful and helpful.

3.1. From the Publisher.

This systematic treatment of one of the oldest topics of mathematical analysis - the solution of functional equations - features numerous detailed proofs. Thorough and elementary in its approach, it will benefit upper-level undergraduates as well as graduate students.

The text is divided into two parts: equations for functions of a single variable and equations for functions of several variables. Starting with equations that can be solved by simple substitutions, the first part examines the solution of equations by determining the values of the unknown function on a dense set. It also surveys equations with several unknown functions and methods of reduction to differential and integral equations. The second part begins with simple equations and advances to composite equations, equations with several unknown functions of several variables and reduction to partial differential equations. The text concludes with explorations of vector and matrix equations.

3.2. From the Foreword.

Although the English edition is not a new book, it is more than a translation of J Aczél, Vorlesungen über Funktionalgleichungen und ihre Anwendungen 1961. Much new material has been added to the text and the bibliography is almost twice its original size. Not all new entries date from later than 1960, the year the German edition was finished, but the many new contributions since 1960 reflect the vigorous development of this theory. I would feel gratified if the English edition helps continue this progress.

The principles governing the topics chosen remain essentially unchanged. Although the presentation is more formal (separating theorems and proofs) and generalisations for more abstract structures are often considered, the elementary and ever-developing character of the material is still evident. Topics such as the domains of functional equations and noncontinuous solutions are more stressed in the new edition.

A new feature in the Bibliography, which the reader might find helpful, is that after each item he finds the page-numbers where that work was quoted.

The revisions in this new edition are partly based on experiences of further lectures in Debrecen (Hungary), Gainesville (Florida), Giessen and Cologne (Germany), and Waterloo (Ontario). Warmest thanks go to students and colleagues at all these places for their valuable comments.

The author is grateful to Dr M Kuczma (Katowice), Dr E Vincze (Miskolc), and Dr M A McKiernan, who read the galley proofs and the repros, and made many important suggestions; to the translator and referee for the English text; and to Academic Press for readily accepting changes resulting in improvement.

Last but not least I thank those who have made valuable suggestions and remarks on the German edition and hope they and others will continue this activity in connection with the English edition.

J Aczél
Waterloo, Ontario, Canada
October, 1965

3.2. Review by: R M Redheffer.
Quarterly of Applied Mathematics 27 (4) (1970), 554.

This book is written by one of the leading experts in the field of functional equations and, as might be expected, it is extremely scholarly, informative and interesting. In the Interests of brevity Aczél interprets his title in a somewhat narrow sense, so that difference equations (for example) are not included. Differential equations of the sort that Truesdell takes as a starting point for his theory of special functions are also excluded. The great variety of topics which is nevertheless considered shows that this decision to restrict the scope was wise. Complete proofs are given for nearly all the main theorems, and the book contains applications of great diversity. (For example, functional equations are used to characterise the dot and cross product, and to characterise those force fields that admit a suitable concept of "centre of gravity." The functional equations associated with transmission-line theory are presented, but the author does not alienate his mathematical readers by giving technical details.) The book is, on the whole, elementary, and is written in a pleasant, informal style that enables one to dip into it on almost any page. The bibliography of over 100 pages is enhanced by references to it scattered in footnotes throughout the text. Inevitably there are some omissions in a work of this scope; e.g., the reviewer's discussion of compound Poisson processes by functional equations (1953) is not referenced, though relevant. Aczél also modestly omits his own name from the index of authors. In summary, this book is warmly recommended as a most welcome addition to mathematical literature.

3.3. Review by: Miklós Hosszú.
Mathematical Reviews MR0208210 (34 #8020).

This new edition is far more than a translation of the author's previous book in German [Vorlesungen über Funktionalgleichungen und ihre Anwendungen]. The text is supplemented by much new material. The bibliography is almost twice its original size and contains more than two thousand items, including not only works purely on functional equations but also many applications to other fields. These additions show the vigorous development of the theory of functional equations.

It is not necessary to detail these results here since the original papers in which they were published have been reviewed separately. However, it must be noted that in many cases the author can simplify some of the original proofs and weaken some of the hypotheses. Also, the presentation in this new edition is more formal, theorems and proofs being separated. The method of treatment is essentially unchanged and is quite elementary; however, algebraic generalisations are often considered and the domains of the functional equations and noncontinuous solutions are stressed more.

3.4. Review by: Frank Hahn.
The American Mathematical Monthly 75 (3) (1968), 314.

In his concluding remarks the author states, "The principal task, however, would be to create a systematic theory of functional equations, whose outlines still appear rather nebulous with our present knowledge. It seems that great difficulties exist, merely in drawing from the structure of a functional equation conclusions concerning the structure of its solutions, ..." This remark indicates what the structure of a book of this nature must be. It is an examination of many special cases of functional equations with some applications to geometry, physics, probability, and group theory. In his foreword to the German edition the author points out that in order to limit the scope of the book functional equations are so defined as not to include differential, integral, and integro-differential equations. To limit the techniques used, equations for operators and functionals are not considered. A perusal of the table of contents shows that this leaves an adequate field to study.

The techniques used in this book are elementary in nature relying mainly on the calculus, a small knowledge of measure theory, and mostly ingenuity. The bibliography is extensive (pp. 383-508) and is listed by the years in which the papers appeared (1747-1965). The book is certainly an excellent reference bringing under one cover material which is scattered throughout the literature. It is not meant as a text and there are no exercises. In the concluding remarks and in the body of the text various unsolved problems are mentioned.

3.5. Review by: Preston C Hammer.
Science 155 (10 March 1967).

This book, an updating of the original edition printed in German, is a welcome and important addition to the mathematical literature. The field of functional equations has mainly been dealt with by amateurs - that is, mathematicians who have developed certain special results cognate with applications to other areas.

Although clues to more general results are indicated, the author stays close to real or complex variables as basic variables for the functions involved. The number of variables and of participating functions is required to be finite. Examples of functional equations are:

          $f(x + y) = f(x) + f(y).$                  (1)

          $F[F(x, y), z] = F[x, F(y, x)].$          (2)

The first is called the additivity equation and the second the associativity equation. Considering Eq.2 alone one sees that functional equations, if permitted, go deep into algebraic structure.

The restrictions placed by Aczel on his presentation allow him to start with J d'Alembert, 1747, as having first given a treatment of functional equations. Euler, Cauchy, Legendre, and Gauss each dealt with certain functional equations. However, N H Abel gave the first known general attack on functional equations in a series of four papers published between 1823 and 1827.

The lack of a systematic theory unifying a major portion of what is now known about numerical functional equations lends a rather ad hoc appearance to the treatment. The importance of the particular equations considered is so great that one should not require extremes of generality. Any one working with mathematical analysis would profit by study of this book. Young mathematicians looking for new worlds might well consider the loose ends proposed by Aczel as a place to start.

A most valuable feature of the book is the attempt to provide references to the literature. Although Aczel kindly does not say so, it is obvious that much redundancy exists in the literature, largely because there was no book like the present one to make it less excusable. There is a chronological bibliography by certain years from 1747 to 1965, as well as an author index and a subject index.

The arrangement of the material seems to result from variable classification systems. Thus, chapter 1 is headed "Equations which can be solved by simple substitution," chapter 2 is entitled "Values of the unknown function on a dense set," chapter 3, "Equations with several unknown functions," and chapter 4, "Reduction, general methods, "and so on. In other words, sometimes the methods, sometimes the independent variables, and sometimes the forms of the equations are the dominant motive. This seemingly unnatural arrangement reflects the fact that the theory of functional equations has not yet congealed into the framework of tedious elegance.

The treatment in general is adequate for a graduate student of mathematics, and is accessible on a rather elementary level, in principle. In fact the author does not cater to everyone who might use functional equations. His applications are not to kinematics, to economics, to computing, and to logic, but to mathematical analysis. From the standpoint of the research mathematician, the book is a compendium of useful techniques, results, and references. For undergraduate study it needs additional motivation and exercises.

Finally, I consider from my limited viewpoint the question of coverage. It seems to me that inequalities might have been used to better advantage. For example, subadditive functions satisfy a functional inequality. So do convex functions. Embedding the equalities deliberately in inequalities even when emphasising the former would have definite advantages in techniques, proofs, and generality. Next, the general principles are often obscured in techniques. I hold to the opinion that a theorem or result is best presented in the most general fashion available, provided this does not generate an unbearable degree of complexity in the conditions. Much of the theory presented holds in a con-text of partially ordered semigroups or groups. I have discussed several Boolean functional equations which are analogous to certain ones considered here. Such functional equations, implicit in definitions of topology, for example, are not discussed. However, on the grounds he has chosen, those of classical analysis, the author has done a thorough job.

4.1. From the Publisher.

On Applications and Theory of Functional Equations focuses on the principles and advancement of numerical approaches used in functional equations. The publication first offers information on the history of functional equations, noting that the research on functional equations originated in problems related to applied mathematics. The text also highlights the influence of J d'Alembert, S D Poisson, E Picard, and A L Cauchy in promoting the processes of numerical analyses involving functional equations. The role of vectors in solving functional equations is also noted. The book ponders on the international Fifth Annual Meeting on Functional Equations, held in Waterloo, Ontario, Canada on April 24-30, 1967. The meeting gathered participants from America, Asia, Australia, and Europe. One of the topics presented at the meeting focuses on the survey of materials dealing with the progress of approaches in the processes and methodologies involved in solving problems dealing with functional equations. The influence, works, and contributions of A L Cauchy, G Darboux, and G S Young to the field are also underscored. The publication is a valuable reference for readers interested in functional equations.

4.2. From the Preface.

This booklet consists of two articles (On applications and theory of functional equations, and International meeting on functional equations - What are they anyway?). The first originated in lecture notes of talks and short courses given at approximately twenty universities of some fifteen states and countries to students and professors. The second was originally written (with more introductory explanations) for high-school students. This might give an idea of the rather broad spectrum of elementary mathematics which is treated here from an advanced point of view.

There are a few topics common in the two articles but even there the methods are somewhat different and complementary, and it is hoped that the two together can serve effectively as introduction to functional equations and preparation for example to the reading of the monographs and encyclopaedic articles mentioned in section 8 of the second article.

5.1. From the Preface.

In this book we shall deal with measures of information (the most important ones being called entropies), their properties, and, reciprocally, with questions concerning which of these properties determine known measures of information, and which are the most general formulas satisfying reasonable requirements on practical measures of information. To the best of our knowledge, this is the first book investigating this subject in depth. While applications, for instance in logical games and in coding theory, will be mentioned occasionally, we aim at giving sound foundations for these applications, for which there are several books available. We try to avoid, however, dealing with information measures that seem to have been introduced just for their own sakes, with no real applications and without really useful properties which would give hope for such applications in the future. (Of course, our judgments may occasionally prove to be inaccurate on both counts.)

No completeness is claimed, neither in the listing of references nor in the book itself. In particular, measures of information for continuous distributions and the theory of information without probability are not included in the text. For the former, we refer to the books by Kullback (1959) and Csiszár (1975); for the latter, to the papers mentioned in the Introduction and to the publications by Ingarden (1963, 1965), Forte (1967, 1970), Forte and Benvenuti (1969), Benvenuti (1969), Kampé de Fériet (1969, 1970, 1973), Kampé de Fériet and Benvenuti (1969), Kampé de Fériet et al. (1969), Schweizer and Sklar (1969, 1971), Baiocchi (1970), and Divari and Pandolfi (1970), et al., and to the comprehensive works by Forte (1969) and Kampé de Fériet (1969, 1973).
...

Not many prerequisites are needed to understand the present book. Some basic calculus, measure, probability, and set theory facts, which can be found in almost all textbooks on these subjects, are all we need. Another discipline upon which we rely heavily is the theory of functional equations, but we intend to present in this text almost everything that we use from this theory, mostly in Sections 0.3 and 0.4. Additional information can, however, be obtained from the book by Aczél (1966). Most of the other notations, expressions, and definitions to be used later are introduced in Sections 1.1 and 2.1; a few are also introduced in Sections 0.2, 1.5, 2.2, 2.3, 3.1, 5.2, and 6.2. Otherwise, many sections can be read independently from one another, and even Sections 0.3, 0.4, 2.3, 3.1, and others may be consulted only when reference to them occurs.

Formulas, definitions, properties, and theorems (propositions, lemmata, corollaries) are numbered consecutively within the sections. For the sake of readers who do not start at the beginning of the book, or who may have forgotten some definitions, names of properties, etc., we refer back to these items quite often. If the reader remembers them, he can simply skip such references.

The first author is responsible for the present form of the book, but extensive use is made of results, methods, and manuscripts by the second author. Thanks are due to participants in courses given in Canada, Hungary, Australia, Germany, Italy, and the United States on the subject, and to past and present co-workers, in particular to Dr C T Ng for remarks on the mathematical content, and to Drs J A Baker, S Burris, L L Campbell, and R D Luce for suggestions aimed at improving the style. ...

5.2. Table of Contents.

Chapter 0, Introduction. Entropy of a single event. Functional equations; Chapter 1, Shannon's measure of information; Chapter 2, Some desirable properties of entropies and their correlations. The Hinčin and Faddeev characterizations of Shannon's entropy; Chapter 3, The fundamental equation of information; Chapter 4, Further characterizations of the Shannon entropy; Chapter 5, Rényi entropies; Chapter 6, Generalized information functions; Chapter 7, Further measures of information.

5.3. From the Introduction.

It is not a particularly surprising statement [its information-measure is not great] that the concept of information proved to be very important and very universal. These days everything from telephones, business management, and language to computers and cybernetics falls under the name. "Information Processing" (for instance the world organisation of computer scientists is the International Federation for Information Processing). On the other hand there are several applications to mathematical subjects and (see, e.g., Jaglom and Jaglom, 1957) even to logical games and puzzles (often found in magazines alongside crossword puzzles) of methods of information theory, in particular measures of information.

The universality and importance of the concept of information could be compared only with that of energy. It is interesting to compare these two (cf. Rényi, 1960). One is tempted to say that the great inventions of civilisation serve either to transform, store and transmit energy (fire, mechanisms like wheels, use of water and wind energy, for instance, for sailing or in mills, steam engines, use of electric, later nuclear energies, rockets, etc.) or they serve to transform, store and transmit information (speech, writing, drum-signals and fire-signals, printing, telegraph, photograph, telephone, radio, phonograph, film, television, computers, etc.). The analogy goes further. It took a long time (until the middle of the nineteenth century) for the abstract concept of energy to be developed, i.e. for it to be recognised that mechanical energy, heat, chemical energy, electricity, atomic energy, and so on, are different forms of the same substance and that they can be compared, measured with a common measure. What, in fact, remains from the concept of energy, if we disregard its forms of apparition, is its quantity, measure, which was introduced some 125 years ago. In connection with the concept of information, this essentially happened a century later, with the works of Shannon (1948). [There is even a "principle of conservation of information" - like that of energy; see Katona and Tusnády (1967) and Csiszár el al. (1969).] Again, if we disregard the different contents (meanings) of information, what remains is its quantity, measure.

As mentioned before, after isolated works by Hartley (1928) (who introduced the entropy of a distribution of events which could be presumed to be equally probable) and others, it was Shannon (1948, Shannon and Weaver, 1949; cf. also Wiener, 1948, in particular for information obtained from a single event) who introduced a measure of information or entropy of general finite complete probability distributions. Shannon (1948) has originally given a characterisation theorem of the entropy introduced by him. A more general and exact one is due to Hincin (1953), generalised by Faddeev (1956). Forte (1973; cf. Aczé1 et al., 1974) gave an acute characterisation.

Schützenberger (1954) and Kullback (1959) have introduced a type (a set of continuum cardinality) of other measures of information, of which Shannon's entropy is a limiting case. Renyi (1960) has introduced similar entropies for possibly incomplete probability distributions and has formulated the problem of characterising all these new entropies. This problem was solved by the authors of this book (Aczél and Daróczy, 1963; Daróczy, 1963, 1964; Aczél, 1964).

For further works in this subject we refer the reader to the References at the end of the book; here we mention only characterisations directly derived from coding theorems by Campbell (1965, 1966) (cf. Aczél, 1974), the introduction of a fundamental functional equation by Tverberg (1958), Borges (1967), and Daróczy (1969), its generalization by Daróczy (1970) (cf. Havrda and Charvát, 1967 ; Vajda, 1968), and the beginnings of research on information without probability by Ingarden and Urbanik (1961, 1962) and by Kampé de Fériet and Forte (1967).

5.4. Review by: J Kampé de Fériet.
Bull. Amer. Math. Soc. 83 (1977), 192-196.

The purpose of the authors cannot be stated more clearly than in the following lines of the preface (p. XI):

"We shall deal with measures of information (the most important ones being called entropies), their properties, and, reciprocally, with questions concerning which of these properties determine known measures of information, and which are the most general formulas satisfying reasonable requirements on practical measures of information. To the best of our knowledge, this is the first book investigating this subject in depth". In fact, from the 234 pages of the book, only 6 are devoted to simple applications to logical games (pp. 33-38) and 17 to optimal coding (pp. 42-50 and pp. 156-164).

But, as the authors write (p. 29) "the problem is to determine which properties to consider as natural and/or essential". From the beginning they make a choice, which implies consequences of paramount importance: the measure of the information yielded by one event $A$ depends only upon the probability $P(A)$ of this event; due to this choice they restrict themselves to the foundations of the classical information theory, initiated in 1948 by Claude Shannon and Norbert Wiener. Of course this classical theory has proved to be very useful indeed in many branches of science, its greatest success being the foundation and development of communication theory: the fundamental hypothesis means that the amount of information given by a message depends only on its frequency; very unexpected messages give considerable information. But, as has been often pointed, no account of the semantic content of the message, could be taken in this way. Moreover there is a subjective aspect of information, which is entirely out of the scope of the classical theory: the same event does not yield the same amount of information to all the observers. These rather obvious remarks show that the classical information theory deals only with one very important, but particular, aspect of information.
...
The book of J Aczél and Z Daróczy represents the summing-up of a long series of fruitful researches: one has the impression that they have so thoroughly explored the field, that there is little chance for the discovery of really new properties of Shannon's entropy and eventually Rényi's entropy; perhaps this outstanding achievement, discouraging further efforts on the same line, will now stimulate explorations of neighbouring fields, taking account of all the aspects of information out of the scope of the classical theory.