# Clive Kilmister's books

We list below books by Clive Kilmister. Several important works were edited by Kilmister but we have omitted these. We have included some information about one book which he translated. The other books, written or co-written by Kilmister, are listed with various information such as extracts from Prefaces, Introductions, Contents, and reviews.

Before giving the list, let us highlight one particular book which stands out both as unusual and as particularly interesting. This is Eddington's Search for a Fundamental Theory: A Key to the Universe (1994) which combines Kilmister's interests in history, mathematics, physics and Arthur Eddington.

Click on a link below to go to the information about that book.

Special Relativity for Physicists (1958) with Geoff Stephenson

Eddington's statistical theory (1962) with B O J Tupper

Hamiltonian Dynamics (1964)

The Environment of Modern Physics: Study in Relativistic Mechanics (1965)

Rational Mechanics (1966) with John Edmund Reeve

Men of Physics: Sir Arthur Eddington (1966)

Lagrangian Dynamics: An Introduction for Students (1967)

Language, Logic and Mathematics (1967)

Physical Applications of Vectors and Tensors (1969) with Horst Teichmann

The Nature of the Universe (1971)

General Theory of Relativity (1973)

Eddington's Search for a Fundamental Theory: A Key to the Universe (1994)

Combinatorial physics (1995) with Ted Bastin

The origin of discrete particles (2009) with Ted Bastin

1. Special Relativity for Physicists (1958), by Geoff Stephenson and Clive William Kilmister.
1.1. From the Publisher.

This concise account of special relativity is geared toward non-specialists and belongs in the library of anyone interested in the subject and its applications to both classical and modern physics. The treatment takes a historical point of view, without making heavy demands on readers' mathematical abilities; in fact, the theory is developed without the use of tensor calculus, requiring only a working knowledge of three-dimensional vector analysis. Topics include detailed coverage of the Lorentz transformation, including optical and dynamical applications, and applications to modern physics. An excellent bibliography completes this compact, accessible presentation.

1.2. From the Introduction.

It is probably fair to say that at the turn of the nineteenth century an optimistic view was taken of the achievements of theoretical physics. Few problems, so it seemed, remained to be solved. There were, it is true, some details which marred this general picture. For example, the advance of the perihelion of Mercury exceeded the predicted amount by nearly 10%. Attempts to describe the interaction of radiation and matter led to a formula which disagreed with experiment. Certain problems in the optics of moving media were stilt unresolved. But few physicists would have thought at the time that these particular features would require for their explanation a complete revolution in physical ideas. The first two difficulties find their resolution in the general theory of relativity and in the quantum theory, respectively. and we shall not be concerned with them in the present book. The optical problems, however. are resolved by the special theory of relativity which is our immediate concern. Before describing these problems in detail (see §4) we shall consider some of the additional difficulties of pre-relativity physics.

1.3. Review by: A Weinmann.
The Mathematical Gazette 44 (348) (1960), 149.

The main purpose of this book is to give an account of the special theory of relativity without making heavy demands on the mathematical abilities of the reader. This is achieved by not using tensor calculus, although this leads to rather clumsy expressions for the transformation properties of four-vectors and six-vectors and the structure of the equations is not so easily seen. The reader is assumed familiar with three dimensional vector calculus and the extra mathematics required is dictated mainly by each particular application.

After a brief discussion of the difficulties of pre-relativity physics, Lorentz transformations are considered from several points of view and this is followed by an account of their direct consequences. More than half of the book is devoted to applications. These range over a very wide field, including many effects in modern physics for which an understanding of special relativity is required. The Authors have succeeded in most cases in giving a concise account of the principles involved. Many references are given to other treatments or extensions of the theory, and these greatly enhance the value of the book.

This book will be welcomed by anyone who wants to know about the special theory of relativity or its applications without having to understand tensor calculus first.

1.4. Review by: William Wilson.
Science Progress (1933-) 47 (185) (1959), 140-141.

Perhaps the most remarkable feature of this work is the fact that it contains no explicit statement of the special principle of relativity on which Einstein founded his theory. Now one of the most important consequences of this theory is that noticed by Minkowski, namely that space and time constitute a four-dimensional continuum; but the authors refuse to have anything to do with Minkowski's "elegant geometrical interpretation because it would involve the introduction of a four-dimensional space and the use of tensors which are outside the scope of this book." In consequence the physicist, whom the authors presume to teach, is faced with all the complications of four-vectors and six-vectors instead of the very simple tensors of Minkowski's space-time.

On page 1 there is the suggestion that relativistic explanations constitute "a complete revolution in physical ideas." So far from this being the case we find a clear, explicit and correct statement of the principle of relativity (restricted to mechanics and inertial systems) in Newton's Principia (Ed. Ill, p. 20, Corollarium V) and, while writing of Newton, we may add that the authors incorrectly state that his mechanics "was based on the concepts of absolute space and time." Though he did indeed believe in absolute space and time his mechanics is not in fact based on this.

The actual equations given in the book appear indeed to be, all of them, correct; but the enormous numbers of them will surely frighten the physicist. One of the weaknesses of the authors is to be found in their interpretation of relativistic formulae. They say, for instance, on p. 39 that "all moving bodies contract in the direction of motion." This of course is nonsense. The change in length referred to is due to transforming from one system of coordinates to another.

The authors refer to Carl Neumann, who introduced the notion of the Body Alpha, as F E Neumann (in the bibliography). He was in fact the son of Franz Neumann of Königsberg. It may be added that the Body Alpha which they describe on p. 2 is certainly not that described by C Neumann in his paper Über die Principien der Galilei - Newton'schen Theorie.

On p. 77 they give the familiar quantum conditions and ascribe them to Sommerfeld. While it is quite true that Sommerfeld discovered these conditions, they had been discovered somewhat earlier by an Englishman (W Wilson, 1915) as Sommerfeld himself has pointed out.

In conclusion it may be said, notwithstanding the foregoing criticisms, that the book will be useful to the physicist because of the great number of important physical equations and formulae, referring to all parts of the subject, which it contains.
2. Eddington's statistical theory (1962), by Clive William Kilmister and B O J Tupper.
2.1. Review by: A H Taub.
Mathematical Reviews MR0198950 (33 #7100).

The authors state that "the object of this book is the clarification of Eddington's later researches and, in particular, those published in his posthumous work Fundamental theory [1946]". They confine their attention to Chapters I-V of this book, that is, to the part described by Eddington as the statistical theory, for two reasons: (1) this part is, to a very large extent, self-contained and thus gives the authors a convenient starting point in analysing the whole theory, and (2) this part contains the clearest expression of Eddington's underlying ideas. It is not claimed by the authors that the ideas are precisely stated by Eddington. They undertake to clarify Eddington's exposition and attempt to replace his faulty arguments by valid ones. It is stated that the latter lead to more or less similar conclusions.

In many cases the conclusions arrived at by the authors differ from those of Eddington. Thus, for example, in Chapter IV of this book the authors state that Eddington's results on the calculation of the ratio of the masses of the proton and the electron are questionable because of a confusion of "scale-fixed" and "scale-free" characteristics. They then present "an improved form of his argument and show that it leads to loss of the numerical result".

This is not a self-contained book. Except for Chapter 1 and Chapter 8 (the final chapter entitled Conclusions), it is essentially a page-by-page criticism of the first part of Fundamental theory. The reader who is unfamiliar with the latter will be well advised to have a copy of it before him while reading the former. The authors did not attempt to give a definitive account of Eddington's ideas nor do they "believe the time to be ripe for this". They do attempt to show that it is possible for a correct theory of the Eddington type to exist, by their detailed commentary and by first discussing the complexity and ordering of scientific theories and pointing out how Eddington used a variety of theory structures in his arguments (and at times changed structures in the middle of a single argument).

They have clarified Eddington's viewpoint but they have not provided (nor do they claim to) a convincing justification of the results claimed by and for Eddington. In spite of the authors' contribution, this reviewer still finds Fundamental theory "an exceedingly obscure, annoying book in which gems of physical insight are momentarily glimpsed in a thoroughly unworthy setting" (quoted from the authors' preface).
3. Hamiltonian Dynamics (1964), by Clive William Kilmister.
3.1. Review by: F Chorlton.
The Mathematical Gazette 49 (368) (1965), 226.

This slim volume presents the Hamiltonian theory of analytical dynamics in a modern form. Thus, it uses such notions as space configuration and the tensor calculus, and contains many applications to quantum theory (of which no foreknowledge is presumed).
...
Occasional digressions into historic developments will hold the reader's attention. The author echoes Forsyth's scathing attack on "the dogma often called the Principle of Least Action" in his endeavour to "delineate the area of humbug" and uses the occasion to castigate Maupertuis. The book is inspired by the more modern work of H C Lee and by the author's own researches and it should be of great interest to postgraduate students. A useful list of references is appended.

3.2. Review by: D Ter Haar.
Mathematical Reviews MR0168155 (29 #5419).

This booklet is an introduction to analytical mechanics intended, it appears, mainly for mathematicians. Most of the material can be found in other textbooks, but the emphasis and approach are often different from the more usual ones. After a brief introductory chapter reminding the reader of the Newtonian equations of motion, rigid body dynamics, constraints, tensor calculus, and so on, the author discusses in the other chapters the Lagrangian equations, both with the time as a parameter and with the time as an extra coordinate, and the Hamiltonian equations.

In the chapters on Lagrangian theory, Feynman's path integral method is discussed in some detail, a topic not usually found in a text on classical mechanics. On the other hand, in the chapters on Hamiltonian theory one misses a discussion of action and angle variables, of such great importance both in celestial mechanics and in modern plasma physics, and a discussion of classical perturbation theory which is such a delightfully elegant discipline. However, the author may have opted intentionally for a short, sweet account rather than a more comprehensive one. There are various historical notes, of which the longest and most delightful one is the story of Maupertuis' principle.

3.3. Review by: D J Montgomery.
Physics Today 18 (3) (1965), 74-75.

Mathematical physics should of course be called physical mathematics, for there is often precious little physics in it. Dr Kilmister is reader in mathematics at King's College, London, and the mathematical-physics series in which the book appears is edited by Dr G Stephenson of the Department of Mathematics at Imperial College, London. Correctly we surmise that in Hamiltonian Dynamics the physics is pushed into the background, and that we had better think twice before accepting the author's statement that his little book is a self-contained introduction to analytical mechanics with emphasis on the Hamiltonian theory. What the author intends, and indeed the scope and the limitations of his work, are give away on page 1 of the text: "The object of mechanics this differential equation $m\bf{r } = \bf{F}.$" Need we labour the point that physicists view the object of mechanics considerably more generally?

Yet once the restricted scope is accepted, the presentation is to be judged excellent. For American students of physics, however, the work can beat most only a valuable supplement to standard physics texts or lecture notes. It treats carefully much of the mathematics glossed over or ignored completely in the usual treatments. More significantly, it points out sophisticated ways of looking at the mathematical bases of mechanics. And it does so succinctly, profoundly, and freshly.
4. The Environment of Modern Physics: Study in Relativistic Mechanics (1965), by Clive William Kilmister.
4.1. Review by: P T Landsberg.
The Mathematical Gazette 50 (374) (1966), 442-443.

The special theory of relativity comes clothed in many different ways. Sometimes it comes wrapped up in such dense sheets of mathematical formula as to effectively hide its structure; it can also come in the woollier garments occasionally offered by philosophers. The present book offers a new approach, using a minimum of mathematics, and some philosophy. The author is at pains to express his arguments in words rather than symbols, and the result is a readable introduction to the subject. It is no more than an introduction: four-vectors and summation conventions do not occur. A newcomer would thus understand the basic ideas of the theory, but he could not be expected to use them or to apply them effectively after perusal of this book. It is a book one would read "for interest", rather than for systematic instruction.

The novelty claimed for the book lies in its emphasis on the environment and its interaction with the system of interest. The book certainly succeeds admirably as a simple introduction to special relativity. (The author, a reader in Applied Mathematics at King's College, London, is co-author of a more systematic book on the same subject.) It is doubtful, however, if it adds to our understanding of the interaction between a system and its environment. The author certainly tries to keep this question in the foreground, but this effort seems sometimes contrived, and not obviously relevant to the standard and accepted theories which are being expounded. Furthermore, it is doubtful whether any discussion of the relationship between system and environment can be adequate without going into statistical mechanics (where this relation- ship is crucial and leads to quite deep problems). Thus, with reservations as to the title, the book can certainly be warmly recommended.
5. Rational Mechanics (1966), by Clive William Kilmister and John Edmund Reeve.
5.1. Review by: C Truesdell.
Amer. Math. Monthly 74 (6) (1967), 748-749.

There are not many mathematicians who will find this new book clearer than the standard ones, e.g. Osgood's Mechanics for beginners and Whittaker's Analytic Dynamics for the advanced. In fact the authors make the difficulties more difficult than need be. If we can teach set theory to kindergarteners, abstract algebra to freshmen, and functional analysis to seniors, why should the "satisfactory treatment ... based on the theory of continuous media" be "outside the scope of the normal undergraduate syllabus"?

There are many special problems compactly stated, set up, and solved at various points. The main departure from older courses seems to lie in the frank discussions of some difficulties concerning constraints and friction, and in the brief treatment of the theory of impulses as a partly independent discipline. Despite the authors' intention to the contrary, the book remains a loosely connected set of interesting special cases, tastefully selected, very British and closely tied to examinations for an "honours course." It will be a welcome addition to many private and institutional libraries.

5.2. Review by: F Chorlton.
The Mathematical Gazette 51 (378) (1967), 370-371.

This latest book on theoretical mechanics contains chapters on the dynamics of particles and rigid bodies treated vectorially and by the methods of Lagrange. Thus the material is conventional but its detailed treatment, derived partly from the work of Mach, is certainly not so.

From the outset the authors abandon the traditionally historical approach to dynamics, based on the works of Galileo and Newton, in favour of what they claim to be a more modern and logical one. Galvanised no doubt by the rigorous axiomatic approaches now universally adopted in all undergraduate courses in pure mathematics they attempt a similar task in dynamics. Their treatment is based on the idea of influence amongst systems of particles in the sense that the acceleration of any one particle of a given system is influenced by the presence of the others. Thus the notion of gravitation is fundamental to their development. From the axioms that are stated they are able to build up definitions of mass, force and gravitational fields and hence to prove Newton's laws of motion. It is claimed that the more conventional works adopt vague procedures in applying the laws of motion to the solution of problems and that the authors' more rigorous treatment will obviate the difficulties inherent in the traditional modes of application. They hope to present theoretical mechanics not as a collection of nineteenth century tricks but as a unified, beautiful and logical subject having great relevance to the modern trends in mathematics.
...

The book is written for undergraduates taking honours mathematics courses and also for postgraduates interested in theoretical mechanics. Much of the bookwork is relegated to exercises at the end of each chapter and so are some of the more standard problems. It will be interesting to see if this work will trigger-off a revolution in the teaching of applied mathematics in universities and colleges. Perhaps the task of simplifying the presentation of these modern ideas will first have to be undertaken by succeeding authors before this can take effect.
6. Men of Physics: Sir Arthur Eddington (1966), by Clive William Kilmister.
6.1. Review by: G J Whitrow.
The British Journal for the History of Science 4 (1) (1968), 81-82.

This book is in two parts, the first being a critical survey and explanation of Eddington's work by Professor Kilmister and the second a valuable collection of eleven extracts from Eddington's books and papers, all of a highly technical character. Seeing that Eddington was a master of English prose and had a gift for comic writing, it is perhaps a pity that no extract from his more popular works enlivens these austere pages.

For the historian of twentieth-century physics, Eddington is an essential figure and at the same time a highly individual one, by no means typical of his generation. In particular, as Kilmister says, his differences with the conventional workers in quantum mechanics cannot be over-estimated.

Eddington's greatest contribution to physics was his theory of stellar structure, based on his realisation that most stars, even those as dense as the Sun, behave like perfect gases. One of the consequences of his interest in stellar constitution was a now comparatively little-known paper, published in 1925, in which Eddington gave a new proof of Planck's radiation formula. The inclusion of this paper in the present book is particularly welcome. Planck's own derivation of his formula was far from satisfactory, and in 1917 in one of his most remarkable papers Einstein gave a vastly improved method of obtaining it. He invoked both Wien's displacement law and Boltzmann's formula for the relative number of systems in different states of energy, but Eddington showed that it is unnecessary to assume the latter.

Each of the eleven extracts from Eddington's writings is prefaced with enlightening explanatory remarks by the editor. These supplement his own development of Eddington's ideas in the first part of the book, and even if these are still in places not always clear, no blame can be attached to Kilmister. He has done an expert job.

6.2. Review by: Raymond J Seeger.
American Journal of Physics 36 (1968), 71.

Part I, the first 96 pages, gives a critical, historical background of the problems that interested Arthur Stanley Eddington, and then points out his significant contribution in each case. I agree with the author that it represents to some degree "an account of theoretical physics in the first half of the 20th century." Owing to the mathematical sophistication of Eddington himself, the book will probably be most useful to graduate students. Chapter I begins with Eddington's primary interest in astrophysics, which had as a climax his extensive and profound treatment of the internal constitution of the stars. The author points out that "he [Eddington] vastly overestimated the importance of radiation pressure." Chapter II deals with "Quantum Mechanics," "The new quantum theory came in 1926, which was just in time for it to have considerable effect on his thought, although he never became an active worker in the orthodox techniques associated with it to any extent." The author discusses Eddington's derivation of Planck's radiation formula, "interesting as an example of Eddington's approach to physics ... problems are of the kind more naturally associated with mathematics than physics in their underlying spirit." He states that "wave mechanics seems to have been psychologically the more telling theory for his development." "Relativity Theory" is the subject of the third chapter. "His interest in astronomy actually led him to investigate the new theory of gravitation-that of general relativity." "Eddington's work opened this work [Einstein] in the most lucid manner, and most of what he says can still be read today as an excellent introductory text book" ["The Mathematical Theory of Relativity" (1924)]. "The importance of the Einstein universe as a cosmological model was something which impressed Eddington all his life." Eddington himself remarked late in life, "The trouble about unified field theories is that there are so many of them, and all of them are right."

Chapter IV deals with "Algebraic Structures," which Kilmister regards as the principal stumbling block to the new reader of his [Eddington's] books." "The realisation that tensor calculus was not the panacea that he had taken it to be made a prodigious difference to Eddington's work." "It was the algebraic aspect (with respect to Dirac's equation) that fascinated Eddington." The author gives a lengthy explanation of the background material. Chapter V, "The Gulf Between Relativity Theory and Quantum Theory," reminds us that "to his [Eddington's] way of thinking quantum mechanics and general relativity were both approximations." Hence "what was needed, Eddington held, was a problem which was sufficiently simple to be tackled by both approximations." The final chapter, "The Eddington Statistical Theory," suggests that "what one needs is a systematic tool with which to compare theories, to say what they have in common, and in what respects they will differ simply because of their own structure." "The appearance of numbers like 1, 4, 6 and 10 - and at a later stage 60 and 136 - is a characteristic of Eddingtonian theories. Such numbers always arise in comparing different theoretical structures." He concludes, "We are not unaware of the widely held view that this difficulty [explanation of his work] was the result of his simply failing to understand the problems involved, and the method of treatment of modern physics."

Part II gives excerpts from some eleven articles of Eddington, including two that were found as manuscripts after his death. In each instance the author gives a brief introduction to the particular significance of the selection. There is a brief general bibliography and a complete list of published papers by Eddington. The author refers to one group of "more elementary level" books. In view of the tremendous effect that Eddington had upon the general public it is unfortunate that some discussion of the contents of these latter books and of their relation to Eddington's own researches has not been made. In the same connection it is regrettable that so little attention was given to the life story of this truly unique individual.
7. Lagrangian Dynamics: An Introduction for Students (1967), by Clive William Kilmister.
7.1. Review by: Bertha Jeffreys.
The Mathematical Gazette 53 (383) (1969), 106-107.

This is a lively and stimulating account of the Lagrangian method at a fairly elementary level. It is by no means a rehash of older textbooks. The author's purpose is to use the Lagrangian method from the start, and to bring in incidentally topics which in most Dynamics text-books are treated before the chapter on Lagrange's equations. At the beginning of the first chapter it is suggested that the reader 'already familiar with a general method can pass to §1.4.' It could be argued that this is just the kind of reader who will get something from the preceding paragraphs and who will be able to read them critically. For example the exposition of the fundamental problem of the Calculus of Variations makes clear the landmarks but omits the finer points such as the specification of the class of admissible functions. Also it is questionable whether it is wise in §1.1 to give the beginner the idea that a 'trick' is being used.

Emphasis is laid in the preface on the advantage of the use of modem notation. In that used for the numbering of general coordinates the student is introduced to the raising and lowering of suffixes in such a way that he will find himself in the same relation to the use of this technique as M Jourdain did to speaking prose.
...
It is difficult to predict how well the order in which the subject is treated here will go down with a reader unacquainted with the subject and working alone. The ideas may very well be helpful to someone preparing a course of lectures.

7.2. Review by: Enok Palm.
Nordisk Matematisk Tidskrift 16 (4) (1968), 160.

Lagrangian dynamics is a central, classical field in rational mechanics that has been dealt with in countless textbooks. Kilmister's book provides an elementary introduction to the theory, intended for students.

It has unfortunately become a rather vague and messy presentation. In particular, I think the derivation of Lagrange's equations is confusing. For example, here the work of the guiding forces (which in the book is called the inner forces) is omitted on the grounds that these forces are in equilibrium. They can be, but it is not necessary at all. A wheel that rolls on a solid surface is exposed to a guiding force that does not perform any work, but it is of course meaningless to say that the force is in equilibrium.

Today, books are often published that are typically hasty work, but they are usually special books intended for advanced use. As this book is written for students and does not offer advanced material, there is no reason to recommend the book. There are better books on the market.
8. Language, Logic and Mathematics (1967), by Clive William Kilmister.
8.1. From the Preface.

In the present century, studies which have been made in the foundations of mathematics have given us all something to think about in everyday life and in ordinary language. Unfortunately, these studies have often been couched in very technical language, and have required a very sophisticated knowledge of mathematics to appreciate them. The purpose of the present book is to explain just enough mathematics in simple terms for a real understanding of the modern developments.

8.2. Review by: A S G.
Current Science 37 (10) (1968), 299.

This is a publication of the New Science Series under the General Editorship of Sir Graham Sutton. The aim of the Series is to provide authoritative accounts of topics chosen from the wide range of modern science. The books have been written to appeal to intelligent readers who are willing to make conscious effort to understand the thoughts and achievements of specialists in various branches of science physical, biological, social, etc.

In the present century, studies which have been made in the foundations of mathematics have been couched in language too technical to be appreciated by other than the mathematicians themselves. The present book attempts to explain just enough mathematics in simple terms to aid understanding of the modern developments in the foundations, logic and language of mathematics. The topics discussed include Boole's Logic, Cantor's Set Theory, The Logic of Whitehead and Russell, The Decision Problem and Church's System, Hilbert's Metamathematics, Gödel Numbers, Turing's Computing Machines.
9. Physical Applications of Vectors and Tensors (1969), by Horst Teichmann and Clive William Kilmister.
9.1. Note.

This book was written in German by Horst Teichmann and translated into English by Kilmister.

9.2. Review by: Bertha Jeffreys.
The Mathematical Gazette 56 (397) (1972), 254.

The book is the fruit of lectures by the Professor of Applied Physics in Würzburg University and is directed mainly at students of physics, natural sciences and engineering. It is in four sections with headings Vector and Tensor Algebra, Vector Analysis, Tensor Analysis and Functional Analysis. The general plan of all the sections is the same; each consists of an explanation of the mathematical theory, followed by a number of diverse applications. Although in the main the treatment is for three dimensions there are references to extensions to n dimensions. The book must be considered in its context as a course of lectures given in parallel with other courses in which the physical problems are dealt with more fully.
10. The Nature of the Universe (1971), by Clive William Kilmister.
10.1. Review by: E Deeson.
Phys. Bull. 23 (1972), 295-296.

A number of books in this series, the World of Science Library, have appeared previously - authoritative, up to date, readable, superbly illustrated. Prof Kilmister's addition to the family appears in a similar mould: yet it does not quite seem to hit the target as did its predecessors. Possibly the subject is a much broader one than the others (although it is certainly changing no more rapidly); perhaps the reason is that to the author cosmology is, presumably, a hobby for he is a professor of mathematics.

Possibly the fault is the reviewer's. Certainly he did not expect a rather vague history of astronomy followed by only a couple of chapters on modern cosmology. That in effect is what The Nature of the Universe is; it is improved only by some well considered work on relativity. Here it is obvious that the mathematician comes into his own.

Let me not be too scathing however. The volume is as up to date and well illustrated as the others. Although I feel that the opportunity has not entirely been grasped, the result is still very presentable and should interest many readers.

10.2. Review by: Robert L Solomon.
The American Biology Teacher 34 (8) (1972), 483-484.

To be able to treat of cosmology without reference to the intricate mathematics behind it is in itself a difficult feat. This the author has accomplished incredibly well. The book covers the development and evolution of thought on the nature and origin of the universe, from the contributions of the early Greeks to the hypotheses of the present day.

The book should be of general interest to high-school students and college undergraduates. The illustrations are exceptionally good. Some elementary knowledge of classical and modern physics is assumed. The reader is exposed to Kepler's laws, Newtonian mechanics, and relativistic theory without recourse to mathematics, for the most part. The author's analogies in such matters as the "clock paradox" are, although not original, well conceived.

The strongest point the book makes has to do with the constant struggle between cosmologic theory and astronomic observation. The author gives a blow-by-blow account of how such ideas as those of the expanding universe and the steady-state universe have been tested by scientific thought. In addition, he gives some attention to the future of cosmology. Kilmister has also sought to identify the techniques that he believes offer the best chances for a breakthrough in cosmologic thought during the next decade. Virtually no hypothesis has been left uncovered - including some rather bizarre ideas about antimatter.

10.3. Review by: Richard E Berendzen.
Journal of College Science Teaching 2 (1) (1972), 54-55.

Of all of man's intellectual pursuits one of the most arresting and challenging involves analysing the cosmos's origin and structure - the fields of cosmogony and cosmology. Although this book's title does not make the emphasis clear, its basic topic is a historical survey of cosmology, emphasising the last decade.

Written at a level suitable for non-science majors, it is produced with an artistic excellence similar to that of the Scientific American. It is replete with informative diagrams and unusual photographs, including some especially good ones of key cosmologists. Since the subject matter is intriguing to most students in all fields, and since remarkably few excellent books exist on it, this one is a welcome addition.

But Kilmister's book is not without its faults - some awkward examples, several confusing explanations, and a few errors of fact. The author's scheme to illustrate astronomical distances is remarkably unelucidating; the sections on history are overly simplistic; the age given for elliptical galaxies is incorrect; the diagram for retrograde motion is seriously misleading, as is the discussion of Copernicus; the explanation of the use of Cepheid variables for distance determination is wrong. And the author suggests that the ancients ascribed the constellations to match "visions" of creatures in the sky; in fact, they simply named the stars in honour of their gods. (We do similar things today: does the George Washington Bridge look like the President? )

Kilmister, a professor of mathematics at the University of London, presents an extremely up-to-date but curiously balanced view of cosmology. His description is that of the physicist or mathematician rather than astronomer, and he overly emphasises the British contribution. He correctly gives considerable attention to the important work of Fred Hoyle, yet devotes inadequate discussion to the enormously significant observational findings, both radio and optical; he stresses the steady state theory while underemphasising the dramatic significance of the microwave background. Although his unusual discussion of relativity is remarkably lucid, his preoccupation with theory is evidenced by his failure in the chapter, "The Next Decade," even to mention radio interferometry, the field a panel of the National Academy of Sciences has recently noted is yielding phenomenally profound and provocative discoveries.
11. General Theory of Relativity (1973), by Clive William Kilmister.
11.1. From the Publisher.

General Theory of Relativity deals with the general theory of relativity and covers topics ranging from the principle of equivalence and the space-theory of matter to the hypotheses which lie at the bases of geometry, along with the effect of gravitation on the propagation of light. The motion of particles in general relativity theory is also discussed. This book is comprised of 14 chapters and begins with a review of the principle of equivalence, paying particular attention to the question of the existence of inertial frames in Newtonian mechanics. The beginnings and foundations of general relativity are then considered, together with modern developments in the field. Subsequent chapters explore the general notion of multiply extended magnitudes; the space-theory of matter; the effect of gravitation on light propagation; gravitational waves and the motion of particles in general relativity theory; and homogeneity and covariance. An invariant formulation of gravitational radiation theory is also presented. The last three chapters examine continued gravitational contraction, a spinor approach to general relativity, and gravitational red-shift in nuclear resonance. This monograph will be of interest to physicists and mathematicians.

Introduction

Acknowledgments

Part I

Chapter I. The Principle of Equivalence.

Chapter II. The Beginnings of General Relativity.

Chapter III. Modern Developments.

References

Part II

1. On the Hypotheses which Lie at the Bases of Geometry.

2. On the Space-Theory of Matter.

3. On the Effect of Gravitation on the Propagation of Light.

4. The Foundations of General Relativity Theory.

5. On the Motion of Particles in General Relativity Theory.

6. Three Lectures on Relativity Theory.

7. Invariant Formulation of Gravitational Radiation Theory.

8. Gravitational Waves in General Relativity: VII. Waves from Axisymmetric Isolated Systems.

9. On Continued Gravitational Contraction.

10. A Spinor Approach to General Relativity.

11. Gravitational Red-Shift in Nuclear Resonance.

Index

11.3. From the Introduction.

This book must be considered a sequel to the earlier one on special relativity but the difficulties in writing it are of rather a different order from those described in the introduction to that book. General relativity, unlike the special theory, is not a general framework but a specific scientific theory, in fact the best theory of gravitation that we have to date. It is often said to suffer unduly from lack of experimental check or confirmation. This fact, however, is essentially due to the extremely good gravitational theory which we had before, which was that of Newton's. In the situation when one theory is very good it is difficult to distinguish between its predictions and those of an alternative. The main difficulty in writing the book, however, has been to know what to leave out. Every reader will find the choice of papers included extremely idiosyncratic, but this is inevitable when there has been such a sudden rapid advance as has occurred in general relativity since 1945. If anyone feels that their work has been passed over, or unfairly treated in any way, I can only offer my apologies.

11.4. Review by: P T Landsberg.
The Mathematical Gazette 58 (405) (1974), 243.

It is said that the general theory of relativity has a certain beauty in connecting previously unrelated concepts. While this is certainly widely admitted, it is not so often emphasised that it has also rather messy aspects. For the physical insights it provides are limited very seriously by the sheer algebraic complexities involved in passing from a metric or a coordinate system to a solution of the field equations. In this book Professor Kilmister has quite rightly brushed this aspect of the theory under the carpet, and has concentrated instead on the attractive and mathematically more transparent aspects. He has contributed a worthwhile 95-page introduction to the theory, thirty pages of which are devoted to "modern developments". These are bound to be selected according to personal interests (Petrov classification, Bondi's news function, etc.); for example there is little about the Schwarzschild solution and nothing about the Kerr metric or the Gödel universe. However, what he does discuss, he handles in a lively and readable way.

There then follow 11 extracts of papers, among which are translations into English of Einstein's papers on the effect of gravitation on light (1911) and on the foundations of the general theory (1916). Particularly, the latter is a real gem among scientific papers in showing great care of exposition and love of subject - features sometimes missing from more modern work. Papers by Bondi, van der Burg-Metzner, Fock, Penrose and Pirani are also included.

Do these extracts of papers help? This is very much a matter of taste. I found the less accessible papers fascinating and well chosen. The more recent papers are easier to obtain in a library and they are already in English, so that their inclusion in this book fills a less important need, and some of them could have been omitted. While all the papers selected by Professor Kilmister are important, they are not all of help to the beginner: and this is a book for beginners rather than for experts. Very sensibly, the author has not included any material from the last fifteen years. I recommend the book for Kilmister's introduction and the English versions of the two Einstein papers
12. Eddington's Search for a Fundamental Theory: A Key to the Universe (1994), by Clive William Kilmister.
12.1. From the Publisher.

Sir Arthur Eddington, the celebrated astrophysicist, made great strides towards his own 'theory of everything' in his last two books published in 1936 and 1946. Unlike his earlier lucid and authoritative works, these are strangely tentative and obscure - as if he were nervous of the significant advances that he might be making. This 1995 volume examines both how Eddington came to write these uncharacteristic books - in the context of the physics and history of the day - and what value they have to modern physics. The result is an illuminating description of the development of theoretical physics, in the first half of the twentieth century, from a unique point of view: how it affected Eddington's thought. This will provide fascinating reading for scholars in the philosophy of science, theoretical physics, applied mathematics and the history of science.

12.2. From the Preface.

Most physicists have no difficulty in seeing physics as a single subject. Yet this view, which was straightforwardly tenable until the end of the nineteenth century is radically inconsistent with the situation since then. There has been a divorce between the theories of the very small and the large scale. Amongst those worried about this the response has been to search for a 'theory of everything'. This phrase has many closely related connotations and to determine which, if any, is the correct one is an inspiring and useful task, not least for the unexpected by-products. So far, however, it has proved a task without any successful outcome. In this book I draw attention to an alternative. Unnoticed by many today, Eddington in the 1930s made great strides towards a different solution of the enigma. It is half a century since I first succumbed to the Eddingtonian magic - I paraphrase Thomas Mann's phrase to try to do justice to my youthful if uncritical absorption in Relativity Theory of Protons and Electrons, which Eddington had published five years or so earlier, in 1936. I had already enjoyed his authoritative Mathematical Theory of Relativity with no more difficulty than that produced by the complex mathematical techniques which were new to me.

Looking back on it, it surprises me that I could take in without a qualm so many of the unorthodox philosophical views in that book. But Relativity Theory of Protons and Electrons was a different matter. Another clutch of mathematical techniques was not enough to obscure a radically new position. It was very much a book of the 1930s. In the first part of that decade, when the book was gradually coming together, the situation in theoretical physics had become very puzzling. On the one hand, the discoveries of special and general relativity in 1905 and 1915 had been wholly absorbed and largely understood. But understood only within the macroscopic bounds set by the theories themselves. They were confidently expected to explain cosmology in due course, they were known to describe more local astronomical phenomena more closely than Newtonian mechanics and for ordinary mechanics they reduced to Newton's system.

On the other hand, the microscopic world had undergone two revolutions. One was at the turn of the century when Planck and Einstein initiated the so-called old quantum theory, a tool-bag of rules for calculating the frequencies of sharp lines in atomic spectra. The second was in 1925-6 when the manifold confusions into which the old quantum theory had fallen were, it seemed, removed not by one but two initiatives, an algebraic one (Heisenberg, Born, Jordan and Dirac) and an analytic one (Schrödinger). The fact that the two were then proved to be substantially equivalent and so were both called the new quantum theory was held to confirm their rightness.

The puzzling character of physics was that there seemed to be no relation at all between the macroscopic and the microscopic theories, even to the extent that the new quantum theory was consistent with Newtonian mechanics instead of with special relativity (let alone general relativity). At first Eddington was content to see the two sides as simply two alternative ways of looking at the world, wholly independent of each other. He played no active role in developing the new quantum theory. But in 1928 Dirac's publication of his equation for the electron, an equation which was a natural development of the new quantum theory and yet was consistent with special relativity, alerted Eddington to the problem, a problem that he soon came to see as an opportunity. The realisation came in a personally painful way, for the equation contradicted a folk-belief, strongly held by Eddington amongst the majority, that relativity had in its possession a mathematical device (the tensor calculus) which could churn out all possible equations consistent with its tenets. Dirac's equation was not of this form.

Eddington set to work at once in a way that was characteristic; he employed a number of illuminating models to guide his mathematics. The principal such model was Maxwell's electrodynamics and the way in which Maxwell, by combining the hitherto related but separate theories of magnetism and electricity, was led to the prediction of the value of a physical constant (the speed of light in vacuum) from the known scale constants that related electric and magnetic units. This model was then interpreted in an enormously generalised way. The theories to be linked became relativity and quantum theory, separate indeed but scarcely related at all. The much harder task of joining them would, if carried out successfully, bring even more in the way of prizes. As the work went on the prizes took the form of four physical constants, but now of the dimensionless kind, the most striking ones being the so-called fine-structure constant $\Large\frac{e^2}{hc}$, for which Eddington finally settled on the value $\large\frac{1}{137}\normalsize$ and the proton-electron mass ratio $\Large\frac{m_p}{m_e}$ for which he gave 1848. At first Eddington's contemporaries were interested but then scepticism took over. In the absence of any other plausible unification of relativity and quantum theory, for quantum electrodynamics was still far in the future, the problem came to be simply ignored, and Eddington's attempted solution with it.

Meanwhile Eddington had progressed to a new position. The two theories were not to be seen as parts of a single whole, as in the Maxwell model, but as alternative descriptions which could, just occasionally, apply to the same situation and the results compared. Out of such a comparison could come numerical values of physical constants. In 1941 much of this background was unknown to me and I was just fascinated by the book on its own terms. As time went on I learnt more and more of the difficulties and the task became that of reconstructing Eddington's work in such a way as to avoid them. In 1945 Fundamental Theory appeared after Eddington's death. It was a disappointment to me, for it did not seem to address the real obscurities of the earlier book. I spent a good deal of time clearing up the algebraic aspects of the theory but when this had been done the basic ideas were not much clearer. Yet they continued to dazzle: flashes of insight grouped round a frame of numerical results. I kept returning to Eddington's work and puzzling over it between the other enterprises that filled my working life. I owe a debt of great gratitude to Ted Bastin who often gave me guidance and help over Eddington during this forty-five years.

Eventually I realised that the time was passed in which I personally could hope to unravel the whole enigma. Yet the problem tackled by Eddington, if without complete success, the seemingly unbridgeable gap between quantum mechanics and relativity, particularly the general theory, remained, and I write that in full knowledge of the recent attempts at 'quantum gravity' of all kinds. A problem still pursued me but now it was a different one. How could it have come about that Eddington, such a lucid writer both professionally and in popular science, wrote two such obscure books at the end of his life? This book is an attempt to get behind Eddington's printed page to answer that question. I found it needed answers on many levels. The history of science, general history, Eddington's personal circumstances and his peculiar philosophy of science all had their part to play. I had to range over much of the twentieth century physics and over Eddington's other work. There was virtually no documentary evidence to go on beyond his printed papers and books, so the reconstruction of his development has been an imaginative journey and the book an intellectual biography. At the end of it I was excited to find a clear picture emerge of how the two books came about.

The Mystery

This book is an attempt to unravel a mystery about the writing of two scientific books by Sir Arthur Eddington, his Relativity Theory of Protons and Electrons of 1936 and his posthumous Fundamental Theory, published ten years later. It is an appropriate time to attempt this, for nearly half a century has elapsed since Eddington's death. There is also a more important reason for this book. The ideas that Eddington thought were behind his books - and it will become clear to what extent these are truly the ideas behind them - were addressed to one specific problem. In the first half of the twentieth century, and certainly from 1926 onwards, physics became rather absurdly divided into two disparate parts. By 1916 general relativity had built a theory of gravitation on top of the successes of the special theory of 1905. It was highly successful on the large scale, notably in astronomy. This whole way of thinking about the world had nothing in common with that of the quantum theory, one form of which started at the turn of the century. Despite its crudity it had considerable success in explaining small scale phenomena. 1926 saw its replacement by a more refined approach but still one totally at variance with that of general relativity. Almost every assumption behind one approach was inconsistent with the other. Eddington was not the only scientist to be concerned about this ridiculous situation. As time has gone on, however, and physics has remained in the same unsatisfactory state. the consensus has been that it is almost indecent to mention the fact. The importance of another look at Eddington's ideas is that it serves to challenge the consensus.

Both of Eddington's books are of a technical nature, ostensibly addressed to the community of physicists, but here I am addressing the general scientifically-interested public, because the mystery is one of general interest. Accordingly, this is not intended to be a technical book. It tries to answer two questions: 'Why did Eddington come to hold the views that he did ?' at both a historical and personal level and 'Were these views correct?'. The second has a technical component, but is truly a philosophical one. The problem is to unravel the various historical, personal and technical strands that are intertwined. At the same time I have to satisfy the community of physicists on some matters. The material for this is to be found in the numbered notes at the end of each chapter, and the corresponding number in the text shows the relevant place.

The mystery, in more detail, is this: Eddington was an outstanding astronomer of his generation and his The Internal Constitution of the Stars (Eddington 1926) was definitive. Before that his Mathematical Theory of Relativity was instrumental in introducing the general theory to an English-speaking readership. The limpid style and elegance of these two books make them still a joy to read. Eddington's prose style is very distinctive. I am aware of only one comparable stylist of this century, and one whose style is remarkably similar, the economist Sir John Hicks. As a result of his later writings on science and philosophy Eddington became a household name in the 1930s. To the many scientific honours heaped upon him during his life was added the Order of Merit in 1938.

Everything changed, and not only for Eddington, in 1928 when Dirac published his wave equation for the electron, which was consistent with relativity and yet of a different form from any envisaged by the relativists. The new initiative prompted by Dirac's equation was what Eddington needed to make, as he thought, a break-through. Instead of a unified theory which would embrace both relativity and quantum mechanics, there was to be a bridge between them. The bridge would consist of certain problems having a particular kind of simplicity, so that they could be treated by either method. As we shall see later, such an idea depends on the truth of an unusual view of scientific theorising which Eddington had come to hold well before 1928. In the same year he began to publish papers on a new initiative in physics, which he collected together in the first of the two books (Eddington 1936). This book was misunderstood and not very well received. Further papers followed and the other book (Eddington 1946) was more or less finished by his death and published posthumously by Sir Edmund Whittaker.

It cannot be denied that the arguments in both Relativity Theory of Protons and Electrons and Fundamental Theory are very obscure. This intense obscurity, following on from the clarity of the earlier books, is the essence of the Eddington mystery. How did he come to write such books? Was it simply that his powers of reasoning had become confused? That is over simple. But E A Milne reviewed Fundamental Theory for Nature (Milne 1947). He had been an adversary of Eddington in the arguments over cosmology and relativity but he says, very fairly,
It must be recognised that Eddington has invented a mathematical technique. a logic of reasoning and a language of formulation which are as yet strange to most of us, and which no doubt conceal arguments and considerations which would be accepted if less individually expressed.
But there is a sting in the tail when he goes on to say:
... whether or not it will survive as a great scientific work, it is certainly a notable work of art.
That the comment is meant to be critical is underlined by Milne's analysis of Eddington's claim of a second independent derivation of what he calls 'The central formula of unified theory':
Unfortunately for Eddington's aphorism, it emerges with the wrong power of $N$, in fact with an extra unwanted factor $N^{\frac 1 3}$, which makes nonsense of the so-called independent derivation. But Eddington is not one whit abashed. He calmly puts $m_1 = m_0 (\frac 3 4 N)^{\frac 1 2}$ ... But when Eddington had the bit between his teeth, no mere mathematical inconsistency held him back.
Today the majority opinion would probably be that Milne was being unduly generous. Even if you hold Milne's view, the question of why Eddington wrote Fundamental Theory is still relevant. Its answer would be found in psychology perhaps, but it would solve the mystery. On the other hand, if the ideas behind the two books are worth something, as many people still think, can these ideas be brought out into the open? This would dispel the mystery. The two approaches are not mutually exclusive because the matter is more complicated than either suggests. It is this that makes this present book worthwhile.

Something should be said here about Eddington's attitudes to these two main theories of theoretical physics in the twentieth century and also about the criticism of his notion of a bridge between them. In both The Internal Constitution of the Stars and in Mathematical Theory of Relativity Eddington shows his awareness of the simultaneous developments taking place in quantum theory but he takes no part in them. This is particularly striking in the discussion of stellar structure. All the evidence for the discussion of the interior of stars comes from the radiation streaming out of them. This radiation is generated somewhere in the star and it was already clear in the 1920s that some kind of thermonuclear process was involved. It was for this reason that Eddington's great rival in astrophysics, Sir James Jeans, believed that it was too early to expect to understand stellar structure. Eddington's success was to see that the appearance of quantum phenomena could to a large extent be circumvented in the theory. Some problems of stellar evolution would no doubt continue to depend critically on nuclear theory and so became answerable only in the second half of the century, but the main lines could be made clear. The problem in general relativity was less serious. Eddington makes various references in his book to the curious features of quantum laws. On the whole, though, he is content to accept the 'old quantum theory' of before 1926 as a collection of rules of remarkable success since they were understood by nobody. When these rules were replaced by a more extensive theory, the 'new quantum theory' of 1925-6, Eddington saw that some of the new ideas might well be important in general relativity but he did not try to introduce them, for he did not yet see how.

It was not till the initiative inspired by Dirac's equation that Eddington had the notion of a bridge between the theories. Comparing treatments would give numerical values for certain physical constants. Amongst those claimed as a result of calculation by Eddington were the ratio of the masses of the proton and the electron (about 1836) and a constant that measures the strength of the electromagnetic interaction in quantum field theory, known (for historical reasons) as the fine-structure constant (about $\large\frac{1}{137}\normalsize$). The critical comments have tended to concentrate on this notion of calculating numbers that were otherwise thought of as known only experimentally. Eddington is partly to blame for this emphasis. But the comments have tended to ridicule any such theory as impossible and it is hard to see why this should be so. For, firstly, the idea of theoretical physics is to calculate numbers which will then be found by experiment, starting with numbers known already. The difference in Eddington's case was only that the empirical assumptions seemed relatively few.

Secondly, there is a well-accepted example of such a derivation of a numerical constant in electromagnetism. The theories of electricity and magnetism arose separately and each defined appropriate systems of units. Then Maxwell's theory completed the process of unifying the two. But the two sets of units, which had a risen historically, were not the same. A scaling constant was needed to reduce one set of units to the other and it had the dimensions of velocity and a numerical value close to the speed of light. Maxwell's theory predicted the propagation of waves with that speed. The obvious conclusion was that light was one example of electromagnetic radiation. Put it this way: there were two empirically-determined constants, both of the units of velocity. One was the scaling constant between electric and magnetic units; the other was the speed of light. Their ratio was very near to unity and Maxwell's theory then showed it to be exactly unity. Indeed, there is a much earlier analogy to such a calculation, noticed by Sir Edmund Whittaker. That is Archimedes' determination of limits on the value of π by determining the lengths of inscribed and circumscribed polygons. One could see π as a (very basic) physical constant whose value could be determined empirically by rolling a circular disc along a line. Archimedes showed that its value must lie between $\large\frac{22}{7}\normalsize$ and $\large\frac{223}{71}\normalsize$.

A simpler calculation in a more modern spirit than that of Archimedes is provided by considering squares inscribed and circumscribed to a circle of unit diameter. The perimeter of the circumscribing square is then evidently 4 and that of the inscribed square is $4 \times \large\frac{√2}{2}\normalsize = 2.828 ....$ These provide (rather crude) limits to the possible value of $\pi$ and their average, 3.414 ... is an approximation correct to about 8.5%.

Thirdly, such an orthodox physicist as Dirac expressed the opinion that quantum field theory was incomplete precisely because it failed to calculate the value of the constant measuring the strength of electromagnetic interaction, the fine-structure constant, a. It is true that his argument was specific to this one constant. It rested on the fact that the whole procedure of quantum electrodynamics is meaningful only because the value of a is small enough. But even so, this is sufficient to show that the possibility of the calculation of some constants is not to be rejected out of hand .One can only regret that the criticism tended to concentrate on the existence of the calculation of numbers rather than on the obscure arguments by which the numbers were claimed to be found.

12.3. Review by: David Kaiser.
Isis 86 (4) (1995), 675.

In Eddington's Search for a Fundamental Theory, Kilmister sets himself the difficult task of unravelling what he terms the "Eddington mystery": Why were Eddington's books Relativity Theory of Protons and Electrons (Cambridge, 1936) and the posthumously published Fundamental Theory (Cambridge, 1946) so different from other contemporary developments in theoretical physics? Alongside this historical task, Kilmister (a mathematical physicist) pursues an unusual evaluative and resuscitative goal: he tries to sharpen and fortify many of Eddington's original arguments, to see what useful physical insights may be extracted from Eddington's idiosyncratic oeuvre.

Most of Kilmister's study concentrates on Eddington's attempts to calculate algebraically the value of the "fine-structure" constant of quantum theory and the ratio of the proton and electron masses. In both of these cases, Eddington's education in the strong and singular Cambridge tradition of mathematical physics led him to proceed differently from his Continental colleagues. Eddington's efforts to relate these two fundamental constants to obscure quaternion algebras or mysteriously derived quadratic equations often struck other physicists (then and now) as highbrow numerology rather than physical derivations. By retracing Eddington's arguments, Kilmister can at least demonstrate the internal consistencies and motivations behind much of this work, which have not often been appreciated.

Kilmister's exposition is designed to be readable by historians not trained in physics; derivations and further technical discussions are reserved for the footnotes that accompany each chapter. Still, portions of the main text (such as the discussions of quaternion and Clifford algebras) may prove difficult. The sources upon which Kilmister draws are restricted to Eddington's published scientific papers and books, with limited use made of unpublished manuscripts and occasional use made of the published correspondence of other physicists from this period. His study could likely have benefited from more attention to the secondary historical material on the Cambridge training of mathematical physicists and on less unusual developments in quantum theory during this period.

This criticism aside, Kilmister has produced a careful, serious examination of Eddington' s later work; the result is an internal reconstruction and clear exposition of what is notoriously obscure material. By presenting such a close study of this work, Kilmister's book provides an interesting counterstory to the historically well-studied "mainstream" approaches and attitudes of Eddington's contemporary theoretical physicists. In this way, Kilmister's internal reconstruction of Eddington's work shares a common spirit with much recent social history of science. His book can lead to a more "symmetric" treatment of historical disputes, by taking seriously the arguments and positions of the underdog.

12.4. Review by: Elie Zahar.
The British Journal for the Philosophy of Science 48 (1) (1997), 132-139.

Clive Kilmister's book is only partly about Eddington's attempts to unify or bring together Relativity Theory and Quantum Mechanics. It is also a masterly examination of the main themes which dominated the development of twentieth-century physics.
...
The author succeeds in showing that numerological theories - in the modified sense above - need be neither absurd nor even unreasonable. But despite the undoubted genius of thinkers like Kepler and Eddington, numerology - as a programme - appears always to have ended in failure. Putting it in Eddingtonian language: there is a strong suggestion that the 'world-conditions' do not bring forth beings equipped with enough true a priori knowledge to be able to anticipate the workings of nature. Science seems to be in need of continually checking its conjectures against sense-experience.

As already mentioned, this book provides far more than an account of Eddington's search for a fundamental theory. Not only does it contain a panorama, together with a discussion in depth, of the regulative ideas which dominated the development of modem physics; namely: operationalism, subjectivism, falsifiability, conventionalism and the Principles of Identification, Relativity and Quantisation; but the reader can also gain from it an insight into physics which is to be found in no ordinary textbook. Scientific theories are presented as creative responses to concrete problems, not as posits for which problems provide mere illustrations. The reader interested exclusively in the philosophy of science will naturally be able to follow the main text without having to consult the Notes. But with a modicum of mathematical knowledge, one draws maximum profit from reading the text in conjunction or in parallel with the Notes; these ought to be treated as examples and they are provided with so many helpful hints that they constitute a natural and easily accessible complement to the main text. In short: C W Kilmister's Eddington's Search for a Fundamental Theory is a classic which ought strongly to be recommended to all those seriously interested both in physics and in its philosophy.

12.5. Review by: Tim Jordan.
The British Journal for the History of Science 29 (3) (1996), 377-378.

Clive Kilmister takes up the story of Sir Arthur Eddington, a key influence on the introduction of relativity theory to the English-speaking world and most famous as the organiser of the light-bending experiment in 1919 that confirmed Einstein's theory. The puzzle Kilmister takes up is how the widely respected Eddington came to write two books at the end of his life addressing a key problem of physics (the relations between relativity theory and quantum mechanics expressed as the development of a theory of everything), which were generally rejected, even derided.

Exploring this problem involves a wide discussion of physics from the beginning of the twentieth-century to the Second World War. A range of social and intellectual factors seem relevant, turning Eddington's intellectual trajectory into a problem through which the history of modern physics might be examined. Further, Eddington's particular problem, especially articulated as the attempt to provide a theory of everything, provides an interesting historical model for current attempts to provide such a total physical theory. Unfortunately, Kilmister establishes the possibility of this wide historical and philosophical sweep, only to put it to one side in favour of a detailed reconstruction of Eddington's arguments.

This reconstruction also suffers the problem of much historical work on relativity theory and quantum mechanics, because the language of these scientists is essentially mathematical and so expositions of their thought often also become impenetrably mathematical. Kilmister has tried to stay true to the content of Eddington's thought and provide an accessible book by moving the most detailed mathematics to endnotes. Unfortunately, as a former professor of mathematics, Kilmister has a different understanding of accessible mathematics from many. Large parts of this work, not including the avowedly technical endnotes or the two detailed chapters, will be understood only by those with a strong back- ground in mathematics.

If, as a historian, Clive Kilmister demonstrates his competence as a mathematician, he does not buttress his mathematical strength with any great use of existing history of science or any great understanding of the complexity of historical research. Often Kilmister mentions the world beyond Eddington's arguments only tangentially and when he does consider Eddington's fatal illness as crucial it is in a purely internalist manner that does not affect the content of Eddington's work. Kilmister further complicates historical matters by offering his own attempt to repair Eddington's work, as if Eddington were an interesting but wild colleague. This leads to Kilmister offering no historical account of the crucial rejection of Eddington's work because he is unconcerned with 'ill-informed' rejections. In this way, the historical project gives way at important points to physics.

Eddington's Search for a Fundamental Theory is at best old-fashioned internalist history, for those with a strong mathematical disposition and a prior knowledge of the social context of early twentieth-century physics.

12.6. Review by: Jacques Merleau-Ponty.
Revue d'histoire des sciences 50 (1/2) (1997), 225-226.

C W Kilmister was one of the rare mathematicians, who, after Eddington's death in 1944, tried to explore the very original, but very uncertain paths that the great astrophysicist had wanted to open at the end of his life in the search for fundamental laws of the physical world. Fifty years after having, in his own words, "succumbed to Eddingtonian magic", Kilmister seeks to explain the "mystery" of Eddington's late work: how and why the brilliant author of the Internal constitution of the stars (1926), a flagship work of astrophysics, of the Mathematical theory of relativity (1928), a great classic of relativistic thought, he wrote the Relativity theory of protons and electrons (1936), then the Fundamental theory ( posthumous, 1948), works quite unusual in the scientific literature of the century, their obscurity generally recognised as almost impenetrable and practically ignored by the community of physicists - but which nevertheless exert a curious attraction on those who take the trouble to read?

Kilmister's work contains many interesting passages, both from the point of view of philosophy and the history of science; but the complexity of his method of analysis, historical, philosophical and mathematical, if it is indeed Eddingtonian, excludes any attempt at presentation and discussion that is both summary and honest.

Its conclusions are relatively straightforward, however. He notes, certainly rightly, the importance of the surprise that the elaboration by Dirac of the relativistic equation of the electron (1928) was for Eddington: Dirac following a formal and abstract approach of mind that Eddington estimated closer to his than that of most physicists, implemented an algebraic structure different from that which prevailed in General Relativity and which Eddington then believed to be universal in scope - but he thus paved the way, Eddington thought, towards a theory complex enough to complete the unification initiated by Dirac between relativity and quantum theories.

The Relativity Theory of Protons and Electrons (1936) was the result of this new challenge and of the new impetus that Dirac's invention had given to Eddington's thought; very surprising result: the work was "a site rather than an exhibition hall", it presented a thought at work in the form of trial and error rather than a completed system; why then such a hasty publication? Did Eddington, ageing in a threatening world, want, Kilmister wonders, to propose at all costs, if not to impose, a vision of things and of science that was increasingly distant from that which prevailed among his peers?

One of the ideas which emerges from the work must in any case, according to Kilmister be retained: the "Eddington conjecture" according to which the universal numerical constants of physics are not, or not all, contingent data which only observation and experiment can define, but that they must result from the logical-mathematical structure of physical theories and that their values can be calculated a priori; from this conjecture one can, according to Kilmister, draw interesting results, and he himself gives some examples of such research. He also approves of the Eddingtonian idea of a relation to be found between cosmology and the theory of the elements of matter.

The same questions, which arose for the Relativity theory of protons and electrons arise again, but in a way aggravated, for the Fundamental theory, whose incompletion (Eddington died before having completed it) is not sufficient to explain the obscurity. Baffled by the attitude of readers of the Relativity Theory of Protons and Electrons to his attempts to deduce physical constants, Eddington, obstinate and lonely, would have reacted by overriding and offering new deductions of this sort. Encouraged by the success of his popular writings, he also, according to Kilmister, too complacently transposed the brilliant effects of his style into a discourse of strictly scientific intent.

12.7. Review by: Louis H Kauffman.
Mathematical Reviews MR1316891 (96i:81004).

A reconciliation of the very small and the very large scale is one of the most important single issues in physics today. In the 1930's Sir Arthur Eddington, the celebrated astrophysicist, made great strides towards his own "theory of everything". In 1936 and 1946 Eddington's last two books were published [Relativity theory of protons and electrons [1936]Fundamental theory [1946].

This volume examines how Eddington came to write these books - in terms of the physics and history of the day - and what value they have to modern physics. The result is an illuminating description of the development of theoretical physics in the first half of the twentieth century from a unique point of view: how it affected Eddington's thought.

The book is divided into three parts: Part I deals with the period 1882-1928 and the influences of special relativity, general relativity and quantum mechanics on Eddington's thought. Here the book traces the beginnings of Eddington's stance that epistemology is at the basis of physics, that physical laws and physical constants are the consequences of the condition of observation. These roots are in his understanding of general relativity and quantum theory but they are also fuelled by Eddington's shock at the discovery of the Dirac equation. Kilmister shows how Eddington's belief in the ubiquity of tensor representations was dashed by the use of spinors by Dirac. This led Eddington into a fascination with a generalisation of the Dirac algebra and the long search, detailed in Parts II and III, that led to his last two books. Part II contains many remarkable discussions of this algebraic approach and its applications. Part III continues the story in more detail.

This book is enriched by very lucid mathematical notes at the end of each section. It is a startling evocation of a mystery story with two major levels - the level of Eddington's thought (a case closed but not fully grasped) and the level of the mystery of our perception of a physical world with physical laws (a case hardly solved with many detectives still hard at work). Eddington's search for a fundamental theory will be of interest to anyone with a curiosity about physics and the development of theories of nature.
13. Combinatorial physics (1995), by Ted Bastin and Clive William Kilmister.
13.1. From the Publisher.

The authors aim to reinstate a spirit of philosophical enquiry in physics. They abandon the intuitive continuum concepts and build up constructively a combinatorial mathematics of process. This radical change alone makes it possible to calculate the coupling constants of the fundamental fields which - via high energy scattering - are the bridge from the combinatorial world into dynamics. The untenable distinction between what is 'observed', or measured, and what is not, upon which current quantum theory is based, is not needed. If we are to speak of mind, this has to be present - albeit in primitive form - at the most basic level, and not to be dragged in at one arbitrary point to avoid the difficulties about quantum observation. There is a growing literature on information-theoretic models for physics, but hitherto the two disciplines have gone in parallel. In this book they interact vitally.

13.2. From the Preface.

It is nearly fifty years since the authors of this book began a collaboration based on their common interest in the foundations of physics. During that time others have made very major contributions. The fruits of this cooperative enterprise, particularly of the later part of it, are set out here. One problem has led to another. Over that time it became clear that such existing preconceptions as the space and time continua formed an inadequate basis for a physics which has to incorporate a quantum world of discrete character. Here we argue that the impossibility of reconciliation between continuous and discrete starting points means that we must start from the discrete or combinatorial position. Otherwise the quantum theory will remain with confusion and muddle at its centre.

If intuitive clarity is to come from the combinatorial approach it turns out to be hard won because continuum ideas are so deeply embedded in orthodox physics. It has been possible to travel only part of the way but that has been far enough to reveal another positive aspect of the approach. Discreteness is intimately related, according to our theory, to the existence of scale-constants - those dimensionless constants commonly thought by physicists to be of some fundamental significance. We should therefore be able to calculate these. Here we are able to detail the calculation of one, the fine-structure constant, which Paul Dirac emphasised for much of his life as an outstanding problem in the completion of quantum electrodynamics. Our value agrees with the experimentally determined one to better than one part in $10^{5}$. The calculation of dimensionless constants will bring to mind the name of Eddington, and although his work was the original cause of the authors' meeting, and although they agree with him in seeking a combinatorial origin for these constants, their mathematical method, and certainly their calculations of the values of those, have nothing in common with his.

This is a book about physics but philosophers will find that some issues - once their province - which they thought dead and decently buried , are resurrected to new life here. Notable among these is the place of mind. The great originators of the quantum theory knew that the act ion of the mind (or the "observer") had to be part of the theoretical structure, but this development has been aborted. In combinatorial theory there is no escape from the issue. In somewhat the same way, since computing is essentially combinatorial, people in and around computer science may find our representation of physics more natural to them than it is to some orthodox physicists. None the less it is primarily the physics community we seek to inspire to carry further a project of which this is the beginning.

13.3. Review by: Matthew J Donald.
Mathematical Reviews MR1412141 (97j:81007).

Bastin and Kilmister attempt to provide a new approach to the foundations of physics, based on the idea of finite processes. Their criticism of the conceptual framework of standard quantum theory is powerful. As an alternative, they introduce, in several different formulations, a certain combinatorial structure. This structure, which is based on modelling an elementary process of discrimination between differing entities, is a hierarchy of levels. From the hierarchy, Bastin and Kilmister derive numbers which they interpret as determining the relative strengths of fundamental fields. One of these numbers is in close agreement with the experimental value of the fine structure constant. The fact that conventional theory is unable to explain this constant has long been seen as a problem. Unfortunately, most of the empirical successes of conventional theory are themselves left unexplained under Bastin and Kilmister's proposals.
14. The origin of discrete particles (2009), by Ted Bastin and Clive William Kilmister.
14.1. From the Preface.

During the twentieth century the concept of particle underwent a profound change. An electron differs from the particle of Newton in having discrete properties. All electrons have identical mass, charge and spin. The charge and spin of the electron and other elementary particles are logical properties which the particles either have or not. There are a number of other such logical properties. Mass is a more obscure matter but is logical none the less. This book attempts to show why such a change has come about. As a historical matter these investigations did not start from this problem. They began from trying to give physical interpretation to a curious algebraic construction of A F Parker-Rhodes. This construction gave rise to a graded algebra having exactly four levels with which were associated the numbers 3, 10, 137 and $10^{38}$. The rough correspondence of the last two with the inverses of the electromagnetic and gravitational coupling constants and the restriction to four levels suggested a physical connection. It transpired that a very abstract account of the discrete process by which knowledge of the universe is gained provides a reason for an algebraic structure more general than Parker-Rhodes' but having his as a special case. The fact that the more general structure then modifies 137 to a value agreeing with the measured value of the inverse fine-structure constant to better than one part in ten million confirms its correctness. The discrete nature of elementary particles follows from the essentially discrete nature of the process. This explanation is logically prior to one invoking Planck's constant, since assuming the existence of Planck's constant is equivalent to assuming the discreteness. The fact that the algebraic structure contains a substructure agreeing with that of the quarks is a further confirmation of its correctness. Much more work is needed to evaluate all the implications for physics but it is hoped that this book will serve to inspire it.

14.2. Review by: Dean Rickles.
Mathematical Reviews MR2549441 (2011d:81310).

The origin of discrete particles might not unreasonably have been given the subtitle: Down the Eddingtonian rabbit hole! The book develops the programme (to found physics on a combinatorial, process philosophy) that the authors began in their book [Combinatorial physics, 1995] from the same series as the present book. The central evidence is, here again, as in their earlier book, a connection between a mathematical structure (or, rather, a generative hierarchy of structures) and the values of certain important dimensionless constants that the authors think is just too good to be mere coincidence. These constants not only have experimental significance (as the inverses of the electromagnetic and gravitational coupling constants, and also for being responsible for aspects of particle interactions - hence the book's title) but given that there is no other way of deriving them, the authors argue that their successful computation (a computation that occupies them for 46 pages) should compel us to investigate their theory. The logic of their reasoning is this: we have a theory that allows us to compute (with high precision: better than one part in $10^{7}$) otherwise uncomputable physical values; this makes it "very unlikely that [this] success is fortuitous".

This endeavour, they point out, began with an investigation into an algebraic hierarchy discovered by Frederick Parker-Rhodes which gave values close to the (inverses of) the fine structure constant and the gravitational coupling. This is all highly Eddingtonian, of course! However, they give the mathematical structure additional real-world connection by linking it to the process of knowledge acquisition. Also highly Eddingtonian (akin to his "selective subjectivism").

Though this reviewer finds himself unsure about the strength of the theory, it is indeed remarkable how precisely Bastin and Kilmister can compute the value of the fine structure constant - cynically, one might wonder how much of what they wished to get out of their theory went into its construction. We should perhaps pay more attention to such numerical coincidences after Borcherds' proof of the "Moonshine Conjecture". The difference is, of course, that we are being asked to view the numerical matching as evidence for their theory qua physical theory. But whatever one might think of the merits of the theory, the book itself is a fairly edifying read: it is highly original, thought-provoking, and, given its subject matter, surprisingly clear. There are a few moments where one's "crackpot radar" picks up a signal (at mention of the paranormal, for example); but, fortunately, for the most part these are not very intrusive.

Last Updated January 2021