1.1. From the Preface.

This book is designed to serve as a guide to those who would explore the theories by which the scientist seeks to comprehend the mysterious world of the atom. Nuclear fission and atomic bombs are not the whole of atomic science. Behind them lie extraordinary ideas and stirring events without which our understanding would be meagre indeed.

The story of the quantum is the story of a confused and groping search for knowledge conducted by scientists of many lands on a front wider than the world of physics had ever seen before, illumined by flashes of insight, aided by accidents and guesses, and enlivened by coincidences such as one would expect to find only in fiction.

It is a story of turbulent revolution; of the undermining of a complacent physics that had long ruled a limited domain, of a subsequent interregnum predestined for destruction by its own inherent contradictions, and of the tempestuous emergence of a much chastened regime - Quantum Mechanics.

Though quantum mechanics rules newly discovered lands with a firm hand, its victory is not complete. What look like mere scratches on the brilliant surface of its domain reveal themselves as fascinating crevasses betraying the darkness within and luring the intrepid on to new adventure. Nor does quantum mechanics hold undisputed sway but must share dominion with that other rebel, relativity; and though, together, these two theories have led to the most penetrating advances in our search for knowledge, they must yet remain enemies. Their fundamental disagreement will not be resolved until both are subdued by a still more powerful theory which will sweep away our present painfully won fancies concerning such things as space and time, and matter and radiation, and causality. The nature of this theory may only be surmised, but that it will ultimately come is as certain as that our civilization will endure - no more nor less.

What are those potent wraiths we call space and time, without which our universe would be inconceivable? What is that mystic essence, matter, which exists within us and around in so many wondrous forms; which is at once the servant and master of mind, and holds proud rank in the hierarchy of the universe as a primary instrument of divine creation? And what is that swiftest of celestial messengers, radiation, which leaps the empty vastnesses of space with lightning speed?

Though true answers there can be none, science is fated to fret about such problems. It must forever spin tentative theories around them, seeking to entrap therewith some germ of truth upon which to poise its intricate superstructure.

The balance is delicate and every change sends tremors coursing through the edifice to its uttermost tip. The story of relativity tells what happened to science when one provisional theory of space and time yielded to another. The story of the quantum tells of adventures which recently befell our theories of matter and radiation, and of their unexpected consequences.

So abstract a matter as the quantum theory serves well as the basis for learned treatises whose pages overflow with the unfriendly symbols of higher mathematics. Here in this book is its story without mathematics yet without important omission of concept. Here too is a glimpse of the scientific theorist at work, pen and paper his implements, as he experiments with ideas. Not the least of his gifts is a talent for reaching valuable conclusions from what later prove to be faulty premises. For his insight is penetrating. Be it a hint here or a clue there, a crude analogy or a wild guess, he fashions working hypotheses from whatever material is at hand, and, with the divine gift of intuition for guide, courageously follows the faintest will-o'-the-wisp till it show him a way toward truth.

The magnificent rise of the quantum to a dominant position in modern science and philosophy is a story of drama and high adventure often well-nigh incredible. It is a chaotic tale, but amid the apparent chaos one gradually discerns a splendid architecture, each discovery, however seemingly irrelevant or nonsensical, falling cunningly into its appointed place till the whole intricate jigsaw is revealed as one of the major discoveries of the human mind.

1.2. Review by: Leopold Infeld.
The Scientific Monthly 65 (5) (1947), 437-438.

This is an ambitious book. Without being technical, without using mathematics - which is the tool of reasoning and of explanation - the author tries to convey to the reader the fascinating story of quantum theories, including such difficult concepts as phase velocity, the algebra of matrices, Pauli's exclusion principle, the operational calculus, and many others. Comparison, analogies, whimsey - all these devices of popularisation are used. The narrative is often interrupted by interesting historical remarks woven into the story. Another device, which was developed by Jeans into an art of popularisation, is used here, though fortunately in moderation. This device consists in jumping from physics into metaphysics, in creating metaphysical thrills, and in raising the emotions over the "mysteries" of the universe. Thus, we read in Hoffmann's book: "And towering majestically over all, inscrutable and inescapable, is the awful [why awful?] mystery of Existence itself, to confound the mind with an eternal enigma;" or, again, on page 231: "If the mind of a mere Bohr or Einstein astounds us with its power, how may we begin to extol the glory of God who created them?" Many readers will like such metaphysical and religious passages. Those who don't will find enough in the bulk of the book to enjoy it in spite of occasional too-fancy writing. Yet there is one question which, unfortunately, I am unable to answer. Will a reader who does not know anything about quantum theory gain an understanding of the ideas the author tries with considerable skill to explain? My answer to this question is: I hope so, but I am not sure.

1.3. Review by: H S W Massey.
The Mathematical Gazette 32 (302) (1948), 318-319.

The remarkable way in which the conflict between the wave and particle aspects of radiation has been resolved makes a fascinating story but one extremely difficult to make clear even to those with some knowledge of physics. A thoroughgoing attempt to present the subject to the intelligent layman has not been attempted before but Dr Hoffmann has now succeeded in writing a most exciting and, indeed, absorbing story. Assisted by that absence of literary inhibition which characterises American writing, he has portrayed the conflict of ideas between the wave and particle theories in semi-military terms thereby making the whole story read almost as an epic. It is true that for British readers the style may not appeal to the same extent as to transatlantic readers, and a further and perhaps more important criticism is that the uninitiated may gain the impression that scientific research consists of a periodic destruction of all that has gone before and replacement by an entirely different set of "facts", a fallacy all too prevalent among those who have had the misfortune not to have included science in their education. Nevertheless, Dr Hoffmann has not sacrificed accuracy to make a good story and even one familiar with physics would benefit from reading the book. Many of the analogies introduced to make clear some unusual concept are very well chosen and it is often illuminating even to a specialist to realise the relation of an apparently very theoretical idea to everyday occurrences.

Now that Dr Hoffmann has shown that even the complicated study of the emergence and triumph of quantum theory can be described in ordinary language it is to be hoped that other authors will attempt similar popular versions of the progress of theoretical physics. It is not too much to hope that eventually a similar service might be done for those branches of pure mathematics which at present are barely appreciated except by a narrow circle of specialists working almost exclusively in one branch.

1.4. Review by: Homer V Craig.
Mathematics Magazine 22 (2) (1948), 78-80.

Perhaps the most spectacular and serious single experiment in all of recorded history was the test explosion of the first atomic bomb. The attendant publicity, however, was not sufficiently broad to supply a good picture of the genesis of the fundamental ideas. The general nature of the mathematical and experimental activities that have furnished the devastatingly successful modern physical theories has not been adequately disseminated, and certainly there are some unfortunate misconceptions current with regard to scientific discovery. Science, as we know, is not only the most salient feature of our present civilisation, but its potentialities for good and evil are so enormous as to be matters of deep and universal concern. Nevertheless, its recent history has not been awarded sufficient attention, and despite the existence of some excellent books in the popular field, the difficult task of acquainting the general reader with the pertinent facts remains urgent. The general reader has been blessed with many opportunities to learn something of the matters that are relatively easily explained such as elementary mathematics, the lass abstract features of science, the draftman's art, and the "know how" of manufacture and construction; and necessarily these are too often the sole elements of his picture of scientific progress. He has not had an equal opportunity to learn of the role of higher mathematics and quite likely regards higher mathematics as being causally impotent so far as the motivation of scientific discovery is concerned. Certain aspects of radio science, for example, are almost universally known, but how many would ascribe the origin of radio to the mathematical constructions and speculations of James Clerk Maxwell? Fortunately, the present book should prove to be quite effective in restoring the proper balance to the story of scientific progress.

In the reviewers opinion, The Strange Story of the Quantum is a superb piece of expository writing. It bears evidence of unusual linguistic ability combined with a fortunate aptitude for analogy and a good sense of proportion - part of the book is in a light vein while other parts are serious and eloquent; enough of the symbolism and terminology of higher mathematics is present to give a lasting impression of the nature of modern theories without confusing or discouraging the general reader with incomprehensible details. It is made clear that, so far, the secrets of the atom have been wrested from nature by a combination of abstruse mathematics and ingenious physical experiments - that mathematical theory suggests experiment and that the experiments in turn give impetus to the development of the theory. The reader may be surprised to learn that penetration of the unknown is necessarily accompanied by confusion ("It is a poor research worker who insists on fully understanding his own intuition.") and that progress is frequently brought about by accidents. An easy inference from the information presented is that the best way to ensure scientific progress is to encourage research of all kinds. The seemingly impractical may turn out to be devilishly useful.

The general plan of the book is as follows. The prologue presents a description of the famous experiment by Hertz (1887) testing Maxwell's mathematical theory of electricity, magnetism, and light, together with a discussion of various theories of light. Act I, Chapter II is concerned with the birth of the quantum concept and Planck's celebrated radiation formula (1900). Chapter III gives an account of the early contributions of Einstein (1905) to the atomicity of energy. Chapter IV is mainly given over to interference and the photoelectric effect as related to the wave-particle conflict. Chapter V (The Atom of Niels Bohr) touches on radioactivity, Rutherford's concept of the atom, spectroscopy and the Balmer sequence, the incompatibility of the Rutherford atom and Maxwell's theory, and finally the Nicholson-Bohr formula: $\int p dq = nh$. Incidentally, this is the second equation in the book. Chapter VI is concerned with the Zeeman effect, the Stark effect, the Pauli exclusion principle, and the decline of the Bohr theory. Chapter VII is essentially a brief review, while number VIII is a short account of de Broglie's ideas concerning matter waves, and the supporting evidence produced by the Davisson-Germer experiments (electron diffraction patterns). Chapter IX is devoted largely to elementary matrix algebra and Heisenberg's theory. Chapter X introduces the reader very briefly to Dirac's $q$-numbers and the Poisson brackets and gives a discussion of the relation of Dirac's theory to classical physics and the matrix tabulations of Heisenberg. Incidentally, the first eight chapters will probably be more intelligible to the general reader than numbers IX and X. However, this book is never dull and in these as in preceding chapters the author has been able to capture and pass on much of the excitement of the original discoveries. From chapters XI and XII the general reader will learn of the great importance for modern physics of the mathematical work of Sir William Rowan Hamilton, of how Hamilton's work paved the way for the successful Schrödinger wave equation, and finally how Dirac succeeded in displaying the underlying unity in the several apparently diverse theories of the atom. Chapter XIII dwells on some of the conceptual difficulties, such as, Heisenberg's principle of indeterminacy, and the meaning of the symbol in Schrödinger's equation. Perhaps the main lesson of Chapter XIV, which is devoted to metaphysical matters related to the new theories, is contained in the following quotations: "We seem to glimpse an eerie shadow world lying beneath our world of space and time; a weird and cryptic world ..."; "... experiments are a clumsy instrument, afflicted with a fatal indeterminacy which destroys causality. And because our mental images are formed thus clumsily, we may not hope to fashion mental pictures in space and time of what transpires within this deeper world. Abstract mathematics alone may try to paint its likeness." Chapter XV (the epilogue) is taken up for the most part with the contributions of the 1930's - new particles and nuclear fission. "For better or worse, nuclear energy of terrible potency was to be placed in the unready hands of man. ... . The days of the nightmare are upon us ... ." The closing pages contain some miscellaneous comments clothed in language that is at once beautiful and impelling. "Here in such theories and discoveries is a revelation, all too scant of the mighty wonder that is the universe. Here through the minds of our Einstein's and Bohr's we may dimly sense its structural beauty and cunning intricacy, its soaring poetry and its awe-inspiring grandeur and magnificence, with never a hint of its pain and tragic bestiality. ... Now is the terrible crisis of our civilisation. Now is the fateful hour of high decision. For better or worse, We, the People of the Earth, must choose our future. It can be fine and lovable, gentle and dignified, and filled with joy and wonder and thrilling discovery. Or it can be degraded and obscene, despairing and wretched beyond measure, with death and primitive misery stalking the land unchecked." - This is truly an impressive book.

1.5. Review by: F T Adler.
The American Mathematical Monthly 55 (8) (1948), 523-524.

Professor Hoffmann's subtitle gives a concise description of his aim, and of his achievement; he gives an excellent account of the growths of the concepts underlying quantum mechanics and atomic and nuclear theory. Although the author is a mathematician, at present Professor of Mathematics at Queens College, New York, he tells his story, and many of its details, without the use of mathematical formalism. The contents of the theories, their relationship, "parentage," and cross-fertilisation are clearly outlined, so far as it is possible without the use of the mathematical tools. The relations of theory and experiment are stressed at many points.

The book, or, following the author, the drama, starts with Hertz's discovery of electromagnetic waves, that triumph of the predictions of classical theories. Then the "Strange Story" begins to unfold; in Chapter 2 "The Quantum is Conceived an account of Planck's work is given. The photoelectric effect is discussed in Chapter 3; this chapter and the next "Tweedledum and Tweedledee" dealing with interference, bring home, right at the start, the crucial wave-particle dualism. Chapters 4 and 5 give an exposition of the Bohr theory, the Rydberg principle, and the limitations of the Bohr model.

In "Act II" the author proceeds to follow "The Exploits of the Revolutionary Prince." De Broglie's work and the strong impetus of relativity to the development of his thoughts is mentioned in Chapter 8. In Chapter 9, headed "Laundry Lists are Discarded," the author speaks about the Heisenberg-Born Matrix Mechanics, and tries to bring out the mathematical as well as the physical problems involved. This is one of the chapters where this reader had misgivings about the reactions of the "general reader." The following two chapters "The Ascetism of Paul" and "Electrons are Smeared," on the Dirac symbolical method and Schrödinger's wave equation, are high points of the book. In Chapter 12 "Unification" the equivalence of the wave and matrix theories is discussed, and in Chapter 13 "The Strange Denouement" leads to the resolution of the wave-particle paradox, and the role therein of the Heisenberg indeterminacy principle. The last chapter of "Act II," the "New Landscape of Science," describes the general consequences of the new concepts.

In the "Epilogue" the developments of the thirties are traced. The discovery of the new particles, the positron, the neutrino, the neutron, and the meson, and the success of the quantum mechanical theory of coping with them are described.

The book is very good and absorbing reading; it is a serious attempt to show the relationship of mathematics and physics, and to trace the growths and replacement of concepts, without using the mathematical tools of quantum mechanics. It is unfortunate that in so aptly constructed a book the story is a bit overdramatised in places. Sometimes this will not help the reader much, and may lead to misunderstandings, for instance the use of the author's "principle of perversity."

Another point is the lack of units, say on page 22, when speaking of h the author says "Its value is a mere .000 000 000 000 000 000 000 000 006 6." Also one would have liked a stronger emphasis on the permanence of classical theory, as far as macroscopic quantities are concerned. Notwithstanding these few objections, it is a fascinating book, almost in a class with Jeans and Eddington and the other great classics of Popular Science. It will be good collateral reading in courses on the history of science.

2.1. Review by: S Devons.
Science Progress (1933-) 49 (194) (1961), 357-358.

It is difficult for a professional scientist to make an objective appraisal of a popularised presentation of scientific facts and theories with which he is familiar. He should consider not only whether the presentation is intelligible to the layman to whom it is addressed, but also just what sort of impression it is likely to convey, and in judging this latter aspect, opinion may be very subjective.

A book on popular science may be clear, simple, and intelligible, full of useful factual information, but fail to convey any of the excitement or the intellectual atmosphere that a practitioner of the science may associate with the subject. On the other hand attempts to capture the latest spirit and reach the deeper abstractions of science so often make impossible demands on the reader - or perhaps on the author? Only rarely does scientific popularisation combine clarity, simplicity and intelligibility and at the same time capture some of the sense of intellectual excitement and sophistication that lies beyond the usual popular presentation. When this happens the reviewer's task is easy.

Dr Hoffmann's book, first published in 1947, is an outstanding example of popularisation without compromise. It concentrates on a single theme - the development of the basic ideas of quantum physics and this concentration of the energies of the author and the attention of the reader is surely an important factor in its success. But there are other exceptional merits: brilliant use of analogy, vivid and picturesque language, and above all readability. One can read this book from cover to cover more easily than most detective stories, indeed it is difficult to do otherwise.

The 1959 edition contains an additional forty-page postscript which recounts, in the same lively style as the main text, the developments in quantum theory and particle physics of the past decade. Here the topics are more diverse, and the result correspondingly less outstanding than when the author concentrates all his powers of illumination on a single theme.

3.1. From the Publisher.

In this classic critique, a mathematician and educator - who served for many years as a test consultant - challenges the supremacy of standardised testing, demonstrating the inherent flaws in aptitude and achievement tests. Recommended reading for teachers and others involved in education.

3.2. Preface by Jacques Barzun.

For the past thirty months magazines and newspapers have carried a running debate on the theory and practice of testing. By far the greater part of the discussion has consisted of attacks on so-called objective tests, in direct consequence of Mr Banesh Hoffmann articles in Harper's and The American Scholar. Since then academic writers in professional journals have variously said: "I told you so." Doubts and protests long pent up have at last come forth because one man was courageous enough to attack an entrenched position. It is therefore clear that the time is ripe for the full, documented, and reasoned account which Mr. Hoffmann gives in this book of the inadequacies and dangers of mechanical testing.

The vogue of this type of test began after the first world war, during which it had been used by the Army in the hope of rating intelligence and sorting out capacities. Schools and colleges in the 1920's began to give similar tests to their applicants, who, once admitted, were subjected to true-false quizzes instead of the regular essay examinations. Of their own accord, students took whole "batteries" of commercially produced tests to help themselves decide on a career. By the second world war, testing by checkmark was established practice everywhere in American life - in the school system, in business, in the professions, in the administration of law and in the work of hospitals and institutions for the mentally deranged. The production and administration of tests was an industry employing many hard-working and dedicated people.

Half way through this period, in the forties, it was manifestly useless to raise even a question about the value and effect of these tests. When I devoted a short chapter to doing so in Teacher in America, describing with precision enough how mechanical tests raised mediocrity above talent, my remarks were ignored or contemptuously dis missed. I was an obscurantist who lacked the scientific spirit. The most charitable view of my madness was that I was the product of a foreign school system well known to be backward and resistant to modern methods.

Now the tide has turned. As the present book shows, it is the testers who are on the defensive, fighting a rear-guard action against the irresistible force of the argument which says that their questions are in practice often bad and in theory very dangerous. Given the widespread use of tests built on these shaky foundations, their evils affect every literate person, directly or through his children. More abstractly but no less truly, the fate of the nation is affected by what tests do, first, to the powers of those who are learning, and, second, to the selection the tests make among the potential leaders of thought and discoverers of new knowledge. Read Mr. Hoffmann's remarks on the National Merit Scholarship program.

But while American public opinion is recovering from its infatuation with fallacious "methods" in several realms - not only in the giving of tests but also in the teaching of reading, in the training of teachers, in the defining of school subjects, and in the handling of discipline at home - the formerly backward and resistant countries of Europe are zestfully adopting most of our mistakes. England, Germany, France are frolicking with child-centred schools, the permissive system, and the batteries of tests. A recent report from France shows that a long tradition of sobriety is no protection against an attractive error. In the midst of its grave political preoccupations France has been agitated by the discovery, based on tests, of an untutored "genius," a "future Kepler or Galileo" among the eleven children of a modest family in a village near Lyons. The only sceptics about this discovery are the teenage genius himself, his mother, and his sister. "All that people say about me is nonsense," said Jean Ferne to his interviewer, and he followed up this perceptive remark with a description of what he had done to be ranked with the great minds of the past:

"I took the Army tests and did better than average. That gave me a chance for officer training. I took more tests. The seventeenth test was on reading comprehension. They give you a sentence to read, and on one side four others to match, of which two are nearly alike. You put a cross in the right box. I got 17 out of 20. The colonel who gave me the test asked me if I had been guessing. I said my answers seemed to me the most sensible, so he asked me to try again.

"On another test?"

"No, the same. I made the same answers and this time I got 19 out of 20. I wonder why, because I didn't do anything different. The colonel seemed terribly surprised." - (L'Express, March 22, 1962, p. 18)

The colonel is only at the beginning of his own education in these matters. We in the country which originated the game should take care not to be "terribly surprised" at the ease with which self-deception can occur on a national scale. After years of faith in the so-called experiments that proved the validity of the look-and-say method of teaching children how to read, it turns out that the tests (here too) were bad and the results naturally worth less. It is high time to ask what this would-be experimenting in education amounts to. It has long been known in industry that a mere change in the surroundings of production will improve output - temporarily. It is likely that mere change has the same effect in school, and all that the experiments prove is that children respond to novelty in the normal way of increased interest.

With this in mind, the carefully documented recital Mr Hoffmann gives of the way in which the manufacturers of tests defend their product takes on a new importance. For it shows that in contemporary societies the trappings of science are readily used, in good faith, to produce disastrously false results. These results become the stock-in-trade of vested interests. When doubts are uttered, money and prestige are threatened, and indeed all of society is shaken, at least in its easy assumptions. As Mr William Whyte showed in The Organization Man, testing in personnel work does something very different from what was generally thought; and as Mr. Hoffmann shows in the book before us, testing in school and college does the very opposite of what was hoped. In the one case the method represses individuality; in the other it misreads performance.

Every citizen and parent should remember the links in this characteristic chain, which begins with method and ends with gadgetry, whenever proposals come before boards of education to set up large and expensive systems, whether of tests, television courses broadcast from airplanes, or teaching machines. The acts of learning and teaching are more subtle, delicate, elusive, than any method so far found. The desire to teach great numbers does raise difficulties correspondingly great. But it is no solution to do something next door to what is wanted simply because that something is easier to do. If there was not enough milk for growing children would we distribute tap water? Or give them free vaccination against smallpox? Though this is not precisely the analogue of what we have done in the matter of examining the young learner's knowledge, it is precisely true of the arguments used in support of mechanical testing: it is easier and cheaper than the method of confronting mind with mind through the written word.

The further argument that essay examinations cannot be graded uniformly, even by the same reader, only shows again the character of mind itself: it is not an object to be weighed or sampled by volume like a peck of potatoes or a cord of wood. Variations in performance and estimate will always subsist. Hence an objective test of mind is a contradiction in terms, though a fair test, a searching examination, a just estimate, are not. Among the tests that are unfair, certainly, are those which penalise the finer mind - as Mr Hoffmann proves - and those which, through the forceful presence of wrong answers, may divert that mind from the accurate knowledge it possessed a moment before. Anyone who has suddenly doubted the spelling of a word which he was about to write correctly will recognise how easily doubt can work distraction upon thought.

Again, the frequent observation that nowadays the ablest students are the least well prepared (the foolishly called "under-achievers") may well have its source in the neglect of effort which mechanical testing entails. A pupil does not really know what he has learned till he has organised and explained it to someone else. The mere recognition of what is right in someone else's wording is only the beginning of the awareness of truth. As for the writing of essays - and the art of correcting them - excellence can of course not be achieved without steady practice, which, once again, the fatal ease of mechanical testing tends to discourage. But if the tendency of such tests is to denature or misrepresent knowledge, to discourage the right habits of the true student, and to discriminate against the original in favour of the routine mind, of what use are such tests to a nation that has from its beginnings set a high value on instruction and the search for truth? There is no ready answer that is not invidious to the makers of tests. But they too are in good faith, which is why it is urgent and important to study their claims as does Mr Hoffmann, and decide for oneself on which side objectivity does in fact lie.

3.3. From Chapter 1.

On the otherwise unmemorable day, Wednesday, 18 March 1959, The Times of London printed the following letter to the editor:

Sir, - Among the "odd one out" type of questions which my son had to answer for a school entrance examination was: "Which is the odd one out among cricket, football, billiards, and hockey?"

I said billiards because it is the only one played indoors. A colleague says football because it is the only one in which the ball is not struck by an implement. A neighbour says cricket because in all the other games the object is to put the ball into a net; and my son, with the confidence of nine summers, plumps for hockey "because it is the only one that is a girl's game." Could any of your readers put me out of my misery by stating what is the correct answer, and further enlighten me by explaining how questions of this sort prove anything, especially when the scholar has merely to underline the odd one out without giving any reason?

Perhaps there is a remarkable subtlety behind it all. Is the question designed to test what a child of nine may or may not know about billiards - proficiency at which may still be regarded as the sign of a misspent youth?

Yours faithfully, T C Batty.

This question of the four sports makes a fascinating party game. There are many reasons for picking the various answers, and one has only to read the question aloud to start a party off in high gear, with everyone joining in the fun. Any number can play. There is only one drawback: after a while the fun suddenly stops and the party becomes indignantly serious. This happens as soon as someone asks what sense there is in giving children such questions on tests; for then, right away, the fat is in the fire. Parents begin recalling similar questions that their own children had on tests. College students complain that such questions are by no means confined to children. Graduate students and older people push the age limit higher as they recount their own experiences. And soon there is an awed realisation that there may, in fact, be no age limit at all.

But before the party reaches this solemn stage - and before this book does - there is fun to be had. Even the staid London Times could not resist enjoying it. On March 19, the day after the appearance of Mr Batty's letter, it printed the following two letters, in this order, without comment:

... Sir, - "Billiards" is the obvious answer because it is the only one of the games listed which is not a team game. Because the answer is so simple and does not require the child answering it to have a detailed knowledge of the games referred to, I should have thought it a very suitable question for an intelligence test.

Sir, - ... football is the odd one out because ... it is played with an inflated ball as compared with the solid ball used in each of the other three [games].

At this stage I managed to tie myself into an intellectual knot that still has me slightly bewildered. When I had read these three letters it seemed to me that good cases had been made for football and billiards, and that the case for cricket was particularly clever, but that the case for hockey was dubious at best. At first I thought this made hockey easily the worst of the four choices and, in effect, ruled it out. But then I realised that the very fact that hockey was the only one that could be thus ruled out gave it so striking a quality of separateness as to make it an excellent answer after all - perhaps the best.

Fortunately for my peace of mind, it soon occurred to me that hockey is the only one of the four games that is played with a curved implement. But what if I had not thought of that? The problem haunts me still.

In the meantime, the Times had not been idle. On March 20th it published the following letter from an eminent philosopher:

Sir, - Mr T C Batty ... has put his finger on what has long been a matter of great amusement to me. Of the four - cricket, football, billiards, hockey - each is unique in a multitude of respects. For example, billiards is the only one in which the colour of the balls matters, the only one played with more than one ball at once, the only one played on a green cloth and not on a field, the only one whose name has more than eight letters in it. Hockey is the only one ending in a vowel. And so with each of the others.

It seems to me that those who have been responsible for inventing this kind of brain teaser have been ignorant of the elementary philosophical fact that every thing is at once unique and a member of a wider class. Mr Batty's son, in his school class, could be underlined as the only member who was Mr Batty's son. Similarly with every member of his class.

The next day the Times printed a tongue-in-cheek letter of typically British insularity, based as it was on the purely amateur status of hockey in the British Isles:

Quite clearly "hockey" is the correct answer. ... Every child should know that of the four games quoted hockey is the only example of one which, at present, no player is ever paid to play. The examiners are plainly anxious to discover how aware the child is of the problems of choosing a career.

Finally, after a day of meditative silence, the Times grew more serious and printed this letter:

Sir, - In reply to Mr Batty's letter ... I wonder if he would be interested in the results I obtained from a class of 11-year-old children of more than average intelligence and who recently sat for the common entrance examination for secondary education?

I asked them to choose the "odd one out" and then to give their reasons for doing so. The results were as follows: - Football 18, Billiards 17, Hockey 3, Cricket 1.

The reasons they gave were mainly those already submitted by Mr Batty.

I agree with your correspondent and wonder how far this question and questions of a similar nature are a true and reliable guide in the testing of intelligence.

Having enjoyed its romp and returned full circle to the crucial, fun-dampening question, the Times dropped the whole topic from its letter columns. The incident was over, and an opportunity for action had been lost.

But the crucial question is with us still: what sense is there in giving tests in which the candidate just picks answers, and is not allowed to give reasons for his choices?

3.4. Review by: Lyle D Schmidt.
Theory Into Practice 2 (4), Testing and the Schools (1963), 247-248.

Mr Hoffmann's purpose in this book is "to dispel" the awe of professional testing "by subjecting the testers to public examination." The critic "must focus on a particular weak spot in the testers' defences, find a way to turn their favourite weapons into boomerangs, and so cause the testers, in their attempts to defend themselves with improvised weapons, to expose some of their shortcomings to public view."

The author concentrates on a variety of "weak spots" in test development and use. He examines the ambiguity of "objective" questions; the limitations in the instruction to choose the "best" (and not necessarily the "right") answer on multiple-choice tests; the inadequacy of some test titles to describe what the tests actually measure; the misuse of statistical methods; and the inhibiting effect that the prestige of a large test-making organisation has on the potential critics' willingness to speak out.

No one would deny that these are "weak spots" in the development and use of tests. Nor would anyone deny that Mr Hoffmann's analysis of these "weak spots" has aroused defensive reactions among persons who make and use various tests. The book has certainly had the beneficial effect of exposing particular shortcomings of tests. I do not think, however, that these should be the basis for eliminating multiple-choice tests any more than the "weak spots" in Mr Hoffmann's own disciplines of mathematics and physics should lead us to reject them or the constructive contributions they have made.

Despite the potential benefits of exposing shortcomings in testing, the author's manner of attack will, I believe, have a deleterious effect on the proper and constructive uses of tests. His methods are basically journalistic and frequently employ the semantics of propaganda. He regularly refers to the "test psychologists" in a manner that implies they are the "bad guys," contrasting them to the "scholars," who, because they object to these tests, appear to be the "good guys." He makes use of glittering generalities: If the test taker "is strong-minded, nonconformist, unusual, original, or creative as so many of the truly important people are he must stifle his impulses and conform as best he can to the norms that the multiple-choice testers set up in their unimaginative, scientific way." He calls on readers to join the "bandwagon": "This all sounds so reasonable, so logical, and so scientifically conclusive. Yet if we are scholars our instincts rebel. Every step in the argument seems irrefutable. Yet the result feels all wrong." He employs the proof of the single case: "Students are well aware of the short-comings of multiple-choice tests. Here, for example, are excerpts from a letter sent to me by ... a National Merit Scholar." In certain instances, he even fails to document his sources of data, as in the case of a study he reports in which the IQ scores of underprivileged children in an experimental education group were notably increased over a three-year period.

Such techniques may alienate the critical reader who might otherwise have been able to turn Mr Hoffmann's criticisms to good use, either directly (almost every "test psychologist" in the country will read the book) or indirectly. On the other hand, they may appeal to and provide some new arguments for those readers who are out to rout the "professional testers" from their subversive thrones and return the controls of everything to "110-per cent Americans." If this be so, the book will fall far short of achieving its purpose while it might have accomplished so much.

As a matter of fact, with some extensive modification of the manuscript, this book could have made a very worthwhile contribution. In my opinion, it would have been more useful if it had been limited essentially to Chapters 1, 2, and 3, which present some valid and constructive criticisms of test procedures, and Chapter 19, which includes the recommendation for a national committee to look into the purposes and practices of wide-scale testing in the United States. Additional material from some of the remaining chapters would also have been useful if much of the emotional appeal had been eliminated. The energy of its readers might then have been channelled toward the solution of the problems that Mr Hoffmann describes; instead, most of their energy seems to be directed toward a defensive preservation of the status quo ("Hoffmann doesn't know what he's talking about") or toward advocating a giant step backward to "the good old days" ("I told you so!").

3.5. Review by: A Gordon Wilcox.
The Journal of General Education 15 (4) (1964), 301-304.

At midterm in the fall of 1962 a freshman handed in to this reviewer his answer sheet to a required departmental multiple-choice test with the laconic comment "Tilt!" Banesh Hoffmann's The Tyranny of Testing is full of unsettling indications that many serious-minded and conscientious students are no longer content to respond with laconic irony to a testing procedure that unprofessionally conceals its rationale and unscientifically denies the examinee any chance to explain the reasoning behind his answers. The resulting unfairness to superior students is one of the main burdens or motifs of Hoffmann's book, which makes a scrupulously exact and lucidly presented case for "the formation of a distinguished committee of inquiry that will look into the whole matter of testing in the United States from a fresh point of view and form a comprehensive judgment in the public interest." Of the membership of such a committee, "Only a minority should consist of professional test-makers and members of the Boards of test-making and test-giving organisations."

The strategy of Hoffmann's critique is the thoroughly scholarly one of presenting sample test questions for the reader's direct inspection, inviting the reader to try out the questions on others known to be competent in the subject matter of the test, and challenging the testers to show that the questions inspected are not defective. Hoffmann not only lets the reader see for himself that the questions are defective, but quotes the testers in full in their own defence or, as happens more than once, in their own self-condemnation. A high official of Educational Testing Service, for example, said this after being prodded into replying to an examinee's inquiry about an ambiguous test question: "We recognise that there are sometimes questions which present real ambiguities to people with a firm grasp of the principles underlying a question." It is in the artificially produced ambiguity that Hoffmann finds the inescapable flaw of the best tests in the multiple-choice format - a flaw that penalises the very students that a good test should reward: the perceptive students who know enough to see what's wrong with an ambiguous question yet who are denied a chance to reveal what they really do know "no explanations are allowed."

This artificial ambiguity arises from two formal procedural facts about the multiple-choice format: (1) the virtual absence of context, an absence that may at times open even some technical terms to more than one intellectually defensible interpretation, and (2) the fact that the examinees are denied any opportunity to supply interpretive context or otherwise explain the reasons for their answers. Hoffmann also points out the fallacy of assuming redundancy in the sentence-completion procedures: "one does not know the meaning of the sentence as a whole till one knows the whole sentence." That all this does seriously penalise superior students and reward cliché-minded mediocrity is demonstrated in connection with every test question discussed in this book.

Superior students are also penalised by the testers' insensitive disregard of "the rhythm of the context" (see pp. 95-96). Are not Kenneth Pike's studies in intonation a reminder that the rhythm of verbal utterance is as real and decisive a factor in intellection as the lexical meanings of words or their syntax? It is never scientific to be insensitive to distinctions determined by style or rhetoric: the style and rhetoric of irony distinguishes one meaning from its very opposite. For human beings the rhythms of perception and the styles of apprehension are surely as real as the differences between faces.

As a book, The Tyranny of Testing is an exemplar of scrupulously exact scholarship presented with a fine sensitivity to "the rhythm of the context" and to the most rigorous ethical claims scholarship makes. Hoffmann concedes early in the book (and again on the penultimate page) that we have no reason to expect infallibility of any testing method. For this reason he warns us: "Tests are misused when they are taken too seriously. Though testing is no game, people in positions of responsibility would do well to treat it as one." And he urges keeping open "many diverse and noncompeting channels towards recognition." He is fair enough to show how the multiple-choice format was developed in a sincere attempt to solve real problems of excessive subjectivity and excessively variable criteria of competence. He has attacked the best of the test-makers for a constructive purpose: to ascertain whether the demonstrated defects were eradicable or were an inescapable consequence of the format itself.

That the latter is the case is indicated not only by the problem of artificial ambiguity and the procedural denial of explanation, but also by the failure to distinguish between the importance of correct answers and the importance of correct methods, which can be variously fruitful beyond the immediate problem even though a wrong answer is the first arrived at. Here is the crucial issue: are we testing assent or understanding (which may involve competent dissent)? Jacques Barzun states the basic principle in his Foreword: "A pupil does not really know what he has learned till he has organised and explained it to someone else." Nor until this happens does a teacher have the remotest idea of what he has really taught, whether to beginners or to graduate students.

Having built his case on published sample test questions, Hoffmann limits his claim to having made a prima facie case for "the setting up of a distinguished committee of inquiry" whose "minimum concern would be the quality of multiple-choice tests and their makers. The committee should, of course, have free access to confidential tests whose detailed contents may not be aired in public, and it could hardly avoid discussing the complex question of policing tests to ensure that they meet appropriate standards." A prima facie case is not a final verdict but cause for further investigation, which alone can prepare the way for a professionally adequate answer to the fair question posed by the president of the National Merit Scholarship Corporation: "What method of measuring human intellectual attainment and promise, regardless of time or money required, is superior to the tests about which Dr Hoffmann complains?"

These unguarded words about "time and money" recall the principal selling point of the multiple-choice format to academic clerical systems - systems that often function without answer - ability to the inconvenient requirements of scholarly competence and integrity. But efficiency in education does not consist in denying time or avoiding the expenditure of money. Rather it consists in providing adequate time and resources for requisite things to happen (such as students' attaining intelligent understanding) or to be done (such as an unhurried and unharrassed reading of the kind of final exam that allows examinees to demonstrate at inconvenient length but in enlightening detail what they really have learned or why they have come to a given conclusion). It is precisely this inconvenient but competently spent time that our students need. The response to Hoffmann's proposal will indicate how jealous of its professional honour the community of scholars is.

3.6. Review by: Anon.
Federation Bulletin (Federation of State Medical Boards of the United States) 49 (12) (1962), 350-352.

Banesh Hoffman is no stranger to the members of the Federation. He was first brought forcibly to their attention in March 1961 with the publication of "The Tyranny of Multiple Choice Tests" in Harper's Magazine. Comments on this were published in the Federation Bulletin in February 1962 and in May a condensation of Mr Hoffman's article, "Testing" from Physics Today was published.

And now Banesh Hoffman has marshalled all of his arguments against "objective" tests in book form. From the title, "The Tyranny of Testing" one might infer that Mr Hoffman is opposed to all types of tests but one does not have to read far to conclude that he reserves his scorn for the objective types.

In his foreword Jacques Barzun points out that one man has been courageous to attack an entrenched position. He further says, "It is therefore clear that the time is ripe for the full, documented and reasoned account which Mr. Hoffman gives in his book of the inadequacies and dangers of mechanical testing." He further points out that the testers are now on the defensive, "fighting a rear guard action against the irresistible force of the argument which says that their questions are in practice often bad and in theory very dangerous."

Much of the material presented in "The Tyranny of Testing" has appeared in the articles previously mentioned. But considerable new evidence is also presented and the whole argument is well organised. Lest anyone think that Mr Hoffman has something good to say about objective examinations he has only to examine the chapter headings to be disabused. From the first chapter entitled, "A Little Learning Is a Dangerous Thing" to the last involving an outrageous play on words, "Don't Be Pro-Test - Protest," he consistently and clearly expresses his distaste for objective tests. He attacks the professional testers from every conceivable angle and even steadfastly refuses to be impressed by the most impressive weapon of the opposition, statistics.

Mr Hoffman views the present cult of testing with justifiable alarm. He states, "The professional testers are becoming powerful people. With their expertise they are not just crowding out the amateurs but over-awing them. Considering their narrow professionalism, their faith in statistics, their relative indifference to the powerful side effects of their activities, and the enormous impact of these activities on the lives of all of us, we must ask ourselves whether the testers are not already too powerful."

The National Merit Scholarship Corporation does not escape the scorn of Banesh Hoffman. He points to its annual report for 1959 in which it is stated that "about 82 per cent of the scholars rank in the top quarter of their classes even though many have selected colleges of very high academic standing." He further points out that in this year out of 478,971 candidates over 920 actually received merit scholarships. The scholars are certainly a select group. But, although not all of the elite went to colleges of high academic standing, almost 20 per cent of them failed to rank even in the first quarter of their classes. "Do these facts encourage faith in the screening process?"

Banesh Hoffman heaps much of his opprobrium upon the Educational Testing Service for having presumed to test by objective means the facility of the student in English Composition. Oddly enough, Mr. Hoffman believes that the best method of testing the student's ability in composition is to ask him to write. This, of course, shocks the professional testers in that they would be required to read and criticise the compositions written.

Banesh Hoffman's book is of inestimable value to all who are involved in giving examinations. It is of particular interest to members of the Federation who are primarily interested in fitness testing which of necessity revolves around questions involving the fine points of judgement. If nothing else has been accomplished, Mr Hoffman is to be commended for repeatedly pointing out that this is a quality which cannot be measured by machines or statistics.

4.1. Review by: D V Bugg.
Science Progress (1933-) 51 (203) (1963), 506.

The microcosm of atomic physics is strange indeed, far stranger than even the liveliest sceptic of the nineteenth century foresaw. Sometimes atoms and electrons behave like little impenetrable billiard balls, and this most people would be prepared to accept; but at others, they squirm and slither deftly into, around, and even through each other, like pieces of melted chocolate. Yet ask a physicist to explain such peculiar behaviour, and he will explain that it is all basically simple, only you must forget the classical billiard ball picture, and treat atoms with new concepts and a new language, the language of quantum theory.

Banesh Hoffmann tells this story in jaunty fashion in a language and style intended for the layman. Dramatically he describes the conflict between experiments, and the battle between the wave theory and the particle picture which led to the quantum theory. Each historical event and figure, each successive puzzle is highlighted. It is not an easy tale to digest, for quantum theory is a baffling and highly mathematical subject at best, and here it is shorn of all its mathematics. But for the reader who perseveres, there is a well-chosen example or analogy to explain every new twist, every new rule. Scientific readers, too, will find the book amusing, but are likely to miss at times their customary reliance on equations, and may in consequence find parts of the text long-winded.

4.2. Review by: A G H.
CERN Courier 5 (6) (1965), 89-90.

The strange story of the quantum, by Banesh Hoffmann, tells in a lively, readable way the story of the rise of the quantum theory, from its first unseen stirrings in the photoelectric experiments of Hertz in 1887 to its final acceptance as a basic philosophy of science some fifty years later.

Unlike most popular science books, this one does not set out primarily to explain one of the current topics, such as nuclear fission and its uses, but selects instead a history of ideas. As the story is unfolded, explanations take their place in it, and understanding of the ideas - their basic nature if not their intimate detail - comes to the reader in much the same sequence as it came to the physicists at the time.

Also unusual is the fact that this book is written with an emphasis on people, allied to a personification of the various theories involved, which brings the whole thing to life. As with a good detective story, one is led from page to page, anxious to know what will happen next. At the same time, because the author is a mathematician and one-time collaborator of Einstein and Infeld, the story has a ring of authenticity about it. The approach may shock some of the more serious students of the subject, but it will attract many people who would otherwise regard such things as quantum mechanics as being far too mysterious to warrant their attention.

One advantage of a book written in this historical fashion is that it does not get out of date quite so easily as those that attempt to explain the latest technological advances. This one was written originally in 1947 and revised in 1959, yet it does not give the impression of being dated. True it ends in the 'turmoil' of parity non-conservation and remarks that isotopic spin 'seems destined to play a significant role in future developments', there are no muon neutrinos, resonances, $SU_{3}$ or inexplicable $K^{0}_{2}$, decays, but these are only new discoveries in the field in which quantum mechanics is valid; they have not changed its fundamental nature, at least not yet.

The style, and viewpoint of the book are well illustrated by the following extracts, taken from the introductory preface:

'The story of the quantum is the story of a confused and groping search for knowledge ... illumined by flashes of insight, aided by accidents and guesses and enlivened by coincidences such as one would expect to find only in fiction.

'It is a story of turbulent revolution; of the undermining of a complacent physics that had long ruled a limited domain, of a subsequent interregnum predestined for destruction by its own inherent contradictions, and of the tempestuous emergence of a much chastened régime - Quantum Mechanics.' ...

'The magnificent rise of the quantum to a dominant position in science and philosophy is a story of drama and high adventure often well-nigh incredible. It is a chaotic tale, but amid the apparent chaos one gradually discerns a splendid architecture, each discovery, however seemingly irrelevant or nonsensical, falling cunningly into its appointed place till the whole intricate jigsaw is revealed as one of the major discoveries of the human mind.'

This drama of discovery is told in a Prologue, two Acts (divided by an Intermezzo) and an Epilogue, with a Postscript added for the second American edition of 1959 (of which the English edition is a reprint).

On the first page of the Prologue we see Heinrich Hertz at work on the first experiments to detect radio signals, with no thought of wireless telegraphy but just to prove the correctness of Maxwell's electromagnetic equations. And because he was more interested in the phenomenon itself than its practical utilisation he noticed a small effect that was destined to overturn the whole structure of physics. How this first came about is told in the five chapters of Act 1; the 'violet catastrophe', the invention of the energy quantum. Einstein's explanation of the photoelectric effect and the beginnings of the 'wave or particle' controversy, the rise of Niels Bohr's atomic theory and its final collapse as a ruin of contradictions (a fact that will come as a surprise to many readers, and possibly writers, who still believe that the atom really is just a miniature solar system).

After the Intermezzo, which warns of things to come, the next seven chapters, forming Act 2, show how the chaos of the late nineteen twenties gradually resolved itself into a usable theory. Here we find the story of the idea and the experimental proof that electrons were waves as well as particles, the growth of matrix mechanics, the wave equation of Schrödinger and the fundamental formulation of quantum electrodynamics by Dirac, Heisenberg's indeterminacy principle and the abandonment of causality, culminating in a new outlook of physics so different from that of half a century earlier and exemplified by the statement of Dirac that 'the only object of theoretical physics is to calculate results that can be compared with experiment and it is quite unnecessary that any satisfying description of the whole course of the phenomena should be given'.

In the Epilogue we read of the successes of the quantum theory in describing the early experimental results on the physics of the atomic nucleus rather than of the atom as a whole, the discovery of the positron (the first 'antimatter' particle), the neutron, artificial radioactivity, the idea of the neutrino, exchange forces and mesons, fission - and the sudden realisation that all this disinterested, fundamental research, 'science for its own sake', had enormous practical applications.

The Postscript carries the story of the fundamental research further forward into the field of sub-nuclear physics; starting with Powell's discovery of the pion, and stopping for a moment with the shell model of the nucleus, it deals with such things as the fundamental 'infinities' of the wave equations, the Lamb shift and 'virtual particles, renormalisation, Feynman diagrams, strange particles, the overthrow of parity conservation in weak interactions, and ends perhaps on an appropriate topic in view of current events the postulate of CP invariance!

Anyone who doubts whether the happenings of this 'drama" are as important as they are said to be has only to tot up the number of characters who have been rewarded with the Nobel Prize - no less than thirty-seven of them are named, not counting those who appear in the Postscript prior to their award.

Those familiar with the work of CERN and similar high-energy-physics laboratories will also discover many parallels between the events of today and those of the quantum story. There is the same feeling of uncertainty with present theories, the sense of impending discoveries, the theoretical successes (omega minus) and disappointments (intermediate boson); present day physicists have taken over the 'group' algebras of the previous century in the way that the quantum theorists adapted the older mathematics of matrices and the work of Hamilton, for example; and so on.

But Hoffmann's book has an interest for everyone. With some 220 pages of text, in 'pocket' format, it is incredibly good value for money, and for those who wish to use it also for reference it is particularly well indexed.

5.1. From the Preface.

This book is written as much to disturb and annoy as to instruct. Indeed, it seeks to instruct primarily by being disturbing and annoying, and it is often deliberately provocative. If it should cause heated discussion and a re-examination of fundamentals in classroom and mathematics club it will have achieved one of its main purposes.

It is intended as a supplement and corrective to textbooks, and as collateral reading in all courses that deal with vectors. Because the exercises call for no great manipulative skill, and the book avoids using the calculus, it may at first sight seem to be elementary. But it is not. It has something for the beginner, to be sure. But it also has something for quite advanced students and something, too, for their instructors.

I have tried to face awkward questions rather than achieve a spurious simplicity by sweeping them under the rug. To counteract the impression that axioms and definitions are easily come by and that mathematics is a thing of frozen beauty rather than something imperfect and growing, I have mixed pure and applied mathematics and have made the problem of defining vectors a developing, unresolved leitmotif. The book is unconventional, and to describe it further here would be to blunt its intended effect by giving away too much of the plot. A brief word of warning will not be amiss, however. There are no pat answers in this book. I often present ideas in conventional form only to show later that they need modification because of unexpected difficulties, my aim being to induce a healthy scepticism. But too much healthy scepticism can be decidedly unhealthy. The reader should therefore realise that the ideas could have been presented far more hearteningly as a sequence of ever-deepening insights and, thus, of successive mathematical triumphs rather than defeats. If he reads between the lines he will see that, in a significant sense, they are indeed so presented.

To my friends Professors Arthur B Brown and Václav Hlavatý, who read the manuscript, go my warmest thanks. It is impossible to express the depth of my indebtedness to them for their penetrating comments, which have led to major improvements in the text. They should not be held accountable for the views expressed in the book: on some issues I resisted the urgent advice of one or the other of them. A ground-breaking book of this sort is unlikely to be free of debatable views and outright errors, and for all of these I bear the sole responsibility.

5.2. Review by: Seymour Schuster.
The American Mathematical Monthly 74 (7) (1967), 882.

The enormous proliferation of college mathematics textbooks, most of which are uninspired, unoriginal and dull, makes it especially pleasing to pick up an unpretentious paperback like "About Vectors" and find that it does have ideas, genuine originality and a flavour that will give delight to students and instructors alike. Instead of presenting a neat linear path through the subject of vectors, the author raises all sorts of questions which he hopes will "disturb and annoy." The provocative questions and roadblocks strewn in the path of the reader exhibit a wealth of experience in teaching the subject and thoughtful reflection upon the difficulties involved and there seem to be very serious difficulties. For example, early in the game one is faced with the question: What is a vector? Is it any quantity with length and direction? or a directed segment? or an equivalence class? or an undefined algebraic entity? The spirit of raising such questions continues, giving the reader insights into the subject, per se, and to the peculiar ways in which it is applied (in physics). Of particular interest is the chapter on Vector Products and Quotients of Vectors. The embarrassing character of the cross-product is not kept secret; that it is a pseudovector rather than a vector paves the way for an introduction of quarternions and a subsequent chapter on tensors. There is also a very nice discussion relating cross-products to oriented areas and boundaries. This discussion will give students valuable assistance in understanding the integral theorems of Stokes and Gauss. Unfortunately, the chapter on tensors is too bare to provide the same sort of service as its predecessors. No vector calculus appears. This is a mixed blessing, for it makes the book accessible to a large audience but precludes its use as the text for a standard vector analysis course. The principal use for "About Vectors" will probably be as an auxiliary text for any course (including physics courses) in which the subject of vectors plays a central role. However, its use in any manner is sure to give ultimate pleasure in spite of - or because of - the author's design to "disturb and annoy."

5.3. Review by: L D Gates, Jr.
Science, New Series 154 (3753) (1966), 1159.

Vectors are studied in two ways - as elements of abstract vector spaces or as elements of the vector fields of applied mathematics. Banesh Hoffman, in About Vectors, takes the second of these approaches. Vectors are tentatively defined as entities with direction and magnitude and obeying the parallelogram law of addition. During the course of the text the standard operations are introduced after a thorough motivation. The purpose of the book as stated in the preface is "as much to disturb and annoy as to instruct. ... It is intended as a supplement and corrective to textbooks. ..." Specifically, Hoffman calls our attention to awkward questions concerning, for example, free and bound vectors. Some readers will be disappointed that he provides no final resolution to many questions he raises, while others will enjoy the opportunity to clarify their ideas on these matters in their own way. Occasionally Hoffman appears to mystify for no good reason. He observes that forces at different points of a field cannot be added, whereas any pair of vectors in an abstract space can. But this restriction is not found only in vector calculus; any two real numbers can be added, but when calculating values for the sum of functions $f(x)$ and $g(x)$, one does not add $f(2)$ to $g(3)$.

Some features of the book deserve favourable mention. No calculus is used, which makes the book accessible to a wide class of readers. Exercises are scattered liberally through the text and form an integral part. A very interesting section describes vectors which are first associated with surface areas but are found to be more directly related to their rims. On the other hand, it is disappointing that the culminating chapter on tensors contains no examples other than the metrical tensor, although the author is careful to give many applications of vectors in the form of displacements, velocities, forces, and moments.

5.4. Review by: R H Cobb.
The Mathematical Gazette 51 (378) (1967), 331.

The author is Professor of Mathematics at Queens College of the City University of New York. The book, though not very large, is ambitious. It consists of a long sequence of related ideas about vectors, from the first inadequate notions suggested by magnitude and direction in connection with changes in position to the idea of a set of components which transform in a particular way - not that this is presented as the ultimate nature of a vector. There are a few hundred examples, some very short, designed to provoke thought rather than to develop skill in problem solving. References to kinematics, mechanics and geometry show the value of vectors, but the applications are elementary. Indeed, in relation to the traditional syllabus in this country, the techniques required are few. There is no calculus and in Exercise 3.14 of Chapter 4 it is rather strange to see the formula

          $\tan^{-1}\Large\frac {m_1 - m_2}{1 + m_1 m_2 }$

avoided, apparently because it would not be understood; and it is felt necessary to justify the use of radian measure by a comment. This is in strong contrast with the demands made in assimilating basic ideas. The considerable merits of the book lie in the manner in which difficulties in principles, rather than in manipulation, lead to the rethinking of definitions. The topics that arise include location, position vectors, orthogonal triads, the various products, area vectors, axial vectors and quaternions. In the last chapter the subscript and superscript index notation, with its summation convention, is described and the vector is shown in relation to the wider notion of tensors.

This does not pretend to be a text book for use in connection with an examination syllabus; but as a supplement and corrective", to quote the claim in the preface, it is to be commended.

6.1. Note to the Reader.

Every biography is an act of selection, and in the case of a man like Einstein this fact is particularly relevant. Nothing approaching a definitive biography exists. This book does not pretend to be one. We have tried, in brief compass, to give an indication of the man, letting his image come through when possible in terms of his own writings and devoting much space to his science. For science was so much a part of the man, so central to his being, that no biography can be more than anecdotal and superficial that passes over it lightly. Enough is told of the science to show in depth the distinctive character that gave it greatness. Yet, unless the reader is particularly interested in science, he should not pause over subtle details. Our aim has been to present the story in such a way that, by regarding it as pure narrative, the reader can catch the essential flavour of the man and his science, and something of the tumultuous scientific and political era in which he lived and made his extraordinary contributions.

6.2. Extract from the Introduction.

We sketch in this book the story of a profoundly simple man.

The essence of Einstein's profundity lay in his simplicity; and the essence of his science lay in his artistry his phenomenal sense of beauty. "This was sometime a paradox, but now the time gives it proof," as Hamlet said in a different connection.

Already paradox is upon us waiting to be resolved. But there is more to come. As the story unfolds we shall discover that Hamlet's words, thus torn from their context, take on a new and unexpected aptness. For Einstein has strange things to tell about Time.

He is, of course, best known for his theory of relativity, which brought him world fame. But with fame came a form of near-idolatry that Einstein found incomprehensible. To his amazement, he became a living legend, a veritable folk hero, looked upon as an oracle, entertained by royalty, statesmen, and other celebrities, and treated by public and press as if he were a movie star rather than a scientist. When, in Hollywood's glittering heyday, Chaplin took Einstein to the gala opening of his film City Lights, the crowds surged around the limousine as much to gape at Einstein as at Chaplin. Turning in bewilderment to his host, Einstein asked, "What does it mean?" to which the worldly-wise Chaplin bitterly replied, "Nothing."

Though fame brought its inevitable problems, it had no power to spoil Einstein; vanity was no part of him. He showed no trace of pomposity or exaggerated self-importance. Journalists pestered him with irrelevancies and inanities. Painters, sculptors, and photographers, famous and obscure, came in a steady stream to make his portrait. Yet through it all he retained his simplicity and his sense of humour. When a passenger on a train, not recognising him, asked him his occupation, he ruefully replied, "I am an artists' model." Harassed by requests for his autograph, he remarked to friends that autograph hunting was the last vestige of cannibalism: people used to eat people, but now they sought symbolic pieces of them instead. After being lionised at a social affair, he confided dolefully, "When I was young, all I wanted and expected from life was to sit quietly in some corner doing my work without the public paying attention to me. And now see what has become of me."

Long before the public heard of him, Einstein's importance had been recognised by physicists. His theory of relativity has two main parts, the special theory and the general. Not till just after World War I, when eclipse observations lent confirmation to a prediction of the general theory of relativity, did word leak out to the public that something momentous had happened in the world of science.

Einstein came at a time of unprecedented crisis in physics. Relativity was not the only revolutionary scientific development of the early twentieth century. The quantum revolution, which is also part of our story, developed more or less simultaneously and was even more radical than relativity. Yet it made no such public splash and produced no such popular hero as did the latter.

The myth arose that in the whole world only a half-dozen scientists were capable of understanding the general theory of relativity. When Einstein first propounded the theory this may well have been no great exaggeration. But even after dozens of authors had written articles and books explaining the theory, the myth did not die. It has had a long life and traces of it survive even now, when according to a recent estimate the year's output of significant published articles involving the general theory of relativity is somewhere in the neighbourhood of seven hundred to a thousand.

The myth and the eclipse observations gave the theory an aura of mystery and cosmic serenity that must have caught the fancy of a war-weary public eager to forget the guilt and horror of World War I. Yet, even when looked at plain, the theory of relativity remains a towering achievement. In a letter written when he had just turned fifty-one, Einstein indicated that he regarded this theory as his true lifework and said of his other concepts that he looked on them more as Gelegenheitsarbeit work performed as the occasion arose.

But the Gelegenheitsarbeit of an Einstein may not be lightly dismissed. Max Born, who won the Nobel Prize for physics, put it well when he said that Einstein "would be one of the greatest theoretical physicists of all times even if he had not written a single line on relativity." What of Einstein's own Nobel Prize? Suppose we naively take the official citation at face value. Then we may well say that he was awarded the prize primarily for part of his Gelegenheitsarbeit. And all this in no way conflicts with the pre-eminence of his theory of relativity.

Carl Seelig, one of Einstein's chief biographers, once wrote to him asking whether he inherited his scientific gift from his father's side and his musical from his mother's. Einstein replied in all sincerity, "I have no special gift I am only passionately curious. Thus it is not a question of heredity." In saying this, Einstein was not being coy. Rather, he was being careful. He was responding as best he could to an ill-conceived question. If we imagine that it referred to Einstein's scientific artistry, we read into it something that Seelig surely did not have in mind. Implicitly the question put Einstein's music on a par with his science. True, Einstein loved music and played the violin better than many an amateur. But was he, in music, comparable to his favourite composer, Mozart, as in science he was comparable to Newton, whom he revered?

In science Einstein was certainly no amateur. His talents were of thoroughly professional calibre. To the layman the talents of an outstanding professional in any field, whether theology or forgery, can well seem awe-inspiring. But talent is no great rarity, and by professional standards Einstein's scientific talent and technical skill were not spectacular. They were surpassed by those of many a competent practitioner. In this strict sense, then, Einstein indeed had no special scientific gift. What he did have that was special was the magic touch without which even the most passionate curiosity would be ineffectual: he had the authentic magic that transcends logic and distinguishes the genius from the mass of lesser men with greater talent.

This we shall be seeing for ourselves as we proceed. Einstein implicitly conceded it in his autobiography, though in words more consonant with the demands of modesty. After all, he could not, with good grace, say baldly, "I am a genius." This is what he wrote, telling why he became a physicist rather than a mathematician:

The fact that I neglected mathematics to a certain extent had its causes not merely in my stronger interest in science than in mathematics but also in the following strange experience. I saw that mathematics was split up into numerous specialties, each of which could easily absorb the short lifetime granted to us. Consequently I saw myself in the position of Buridan's ass, which was unable to decide upon any specific bundle of hay. This was obviously due to the fact that my intuition was not strong enough in the field of mathematics... In [physics], however, I soon learned to scent out that which was able to lead to fundamentals and to turn aside from everything else, from the multitude of things that clutter up the mind and divert it from the essential.

Such powerful intuition can not be rationally explained. It is not something teachable or reducible to rule, else we might all be geniuses. It wells up spontaneously from within. Albert Einstein wrote his autobiography at the age of sixty-seven, and in it he reminisced about a major event that had occurred more than sixty years before. It is a story that he was fond of telling. Apparently, as a child of four or five, he had been ill in bed and his father had brought him a magnetic compass to play with. Many a child has played with such a toy. But the effect on young Albert was dramatic and prophetic. In his autobiography the ageing Einstein vividly recalled the sense of wonder that had overwhelmed him those many years before: here was a needle, isolated and unreachable, totally enclosed, yet caught in the grip of an invisible urge that made it strive determinedly toward the north. Never mind that the magnetic needle was no more wonderful no less wonderful than a pendulum striving toward the earth. Pendulums and falling objects were already familiar to the child. He took them as a matter of course. He could not realise at the time that they too presented a mystery, nor could he know that later in life he was to make his own great contribution to our understanding, such as it is, of gravitation. To young Albert the magnetic needle came as a revelation. It did not fit. It mocked his early, simple picture of an orderly physical world. In his autobiography he wrote, "I can still remember or at least I believe I can remember that this experience made a deep and abiding impression on me."

These are remarkable words in more ways than one. They tell of the sudden awakening of the passionate curiosity that was to be Einstein's lifelong companion or, it may be, of the sudden crystallisation of something inborn that had already been long in process of formation. Knowing what Einstein accomplished, we can see in these autobiographical words that he found his métier at an early age. Yet there is something strange about his words that will repay our scrutiny. Read them again: "I can still remember or at least I believe I can remember that this experience made a deep and abiding impression on me." Is there not an air of illogic about them? If the experience made a deep and abiding impression on him, surely he should have had no doubts that he remembered its doing so. Why then the precautionary phrase "or at least I believe I can remember"?

Have we caught the great Einstein in a contradiction? Superficially, yes. Yet in a deeper sense we have not. He had told the story often. He knew the frailties of memory. He knew that with repetition a story can become exaggerated, and the teller come to believe it nevertheless. He believed that the magnet had made an unforgettable impression on him. Yet perhaps the effect had not been quite as great as he had come to think. Note how artlessly he conveyed this thought that lay at the back of his mind. The words of caution are unpremeditated. They interrupt the logic. They burst in uninvited, like a Freudian slip, and reveal Einstein's instinctive striving after truth. And they do more. They show us Einstein deepening a truth by means of a paradox.

6.3. Review by: J L Heilbron.
Isis 65 (1) (1974), 128-129.

This popular book offers little to scholars except for a few hitherto unpublished documents whose provenance is not given. Some of the illustrations - for example, the last page of calculations in Einstein's hand and the Herblock tribute at his death - may also be of interest.

Since Banesh Hoffmann once worked with Einstein and had in Helen Dukas, Einstein's long-time secretary, an incomparable source of information, one might expect that his book would contain valuable insights into Einstein's life and character. One would be disappointed. In biographical detail and sensitivity Hoffmann's book compares unfavourably to Ronald Clark's recent Einstein, The Life and Times, which itself has fundamental flaws.

Only in his effort to popularise science does Hoffmann go beyond Clark. His account of special relativity succeeds; elsewhere he runs too quickly to make himself intelligible. "Since the details need not concern us, let the following suffice, even if it seems like mumbo-jumbo." "This inadequate remark will have to suffice, for if we are to keep up with the headlong pace of Einstein's discoveries we must hasten on." One wonders why a man so eager to have done should have begun at all.

6.4. Review by: Edmund G Geiger.
The Science Teacher 40 (6) (1973), 67.

Banesh Hoffmann worked with Einstein at the Institute for Advanced Studies. In writing this book, he collaborated with Einstein's secretary, Helen Dukas, who was associated with him for the last 27 years of his life. Hoffmann seems well qualified to discuss the scientific theories covered in the volume, since he was selected by the British Broadcasting Corporation to explain in layman's language Einstein's theories of relativity.

Although the book emphasises Einstein's personal life, it includes many of his basic scientific developments as well as ideas evolved by other scientists working on similar problems.

The author occasionally discusses Einstein's intuition regarding various aspects of his work. Moreover, he states that it is impossible to visualise as a physical happening the advanced mathematics which is used to state the concepts of relativity, time-space, gravitational effect on light, and related developments. Apparently, Einstein could evolve in difficult mathematical terms what most people require a physical model to understand. Most of the book is easily understood even though certain parts relate to mathematical developments. It contains more than 100 carefully selected illustrations. Junior high and above.

6.5. Review by: Philip Morrison.
Scientific American 228 (3) (1973), 122-124.

"The letter killeth, but the spirit giveth life." This brief, loving, intimate and often poetic volume transmits the spirit of Einstein and his work with a fidelity easily lost in the noise of the enormous documentation that has begun to mask that wry, profound and simple man. Banesh Hoffmann is a mathematician and relativist who worked with Einstein for years in direct and successful collaboration, and Helen Dukas was Einstein's knowing and devoted secretary from the last Berlin years until his death in the hospital in Princeton.

The pair bring two gifts to their readers. The mathematician, an expert pedagogue to the non-mathematical, has constructed a generally quite happy set of simple analogies and parallels to make plain the meaning of the main junctures in Einstein's life of science, whereas his partner offers her close testimony over decades in illumination of what the man meant and felt. Citations are ample and frequent, and the photographs combine the classically familiar with the fresh and unique. Consider Einstein's family and himself in silhouettes he cut out in 1919, or the postcard sent from Paris by two old men - whom we see in an earlier photograph with Einstein when all three friends had been poor students in Bern 50 years before, before the quantum, before the end of simultaneity, before the holocaust, before the bomb.

The book begins with an acute glimpse of Einstein by way of his own autobiographical notes of 1949 (in Albert Einstein: Philosopher Scientist, edited by Paul A Schilpp), an essay so scrupulously intellectual, although both candid and clarifying, that it mentions no private matters at all. Those notes tell of the sense of wonder he felt as a boy when he first saw a magnetic compass, but they do not mention "that his father, who had shown him the compass, was named Hermann." This book at hand is not so austere. It presents a full enough narrative of his life, now and again dwelling on the issues that his mind found central, those that indeed changed our world once and for all.

The text is without algebra; all the explanations are aimed squarely at the lay reader. Although they have limitations, they broadly succeed. Indeed, we see only three pages of algebra, all in holograph reproductions of Einstein's own x-erei. One is lecture notes written in Berlin in 1918, the page marked "Cancelled because of revolution"; one bears on the reverse an unpublished quatrain on Newton in Einstein's German hand; the third, curiously touching, is the last this "hardened x-brother" ever calculated, with long strings of tensor elements firmly laid down by that hand at 76, during his terminal illness. There is painfully hard work at the foundation of the lofty conceptual structures of the century's greatest physicist.

The year of wonder, 1905, is well treated. That was the year of the five papers submitted to the Annalen der Physik, four before July, by a technical examiner in the Patent Office, a young physicist of obviously deep ability. One of the papers, which formed his Ph.D. thesis, derived Avogadro's number from diffusion in liquids; it was the least important, and it appeared in 1906. One earned a Nobel prize; it insisted that the quanta of light suggested by Planck be taken seriously, that these particles made sense out of two fluorescence effects and certain curious experiments of Philipp Lenard (who ended up virulently Nazi) on the ejection of electrons from metal surfaces by light.

That summer Einstein wrote to a friend, a teacher of mathematics in Schaffhausen: "I promise you four papers in exchange. ... The first ... is very revolutionary." One of the other two explained quantitatively the random motion of dust specks under the microscope - the famous Brownian motion - and provided the best evidence for the reality of molecular motion. (Marian von Smoluchowski did the same, perhaps a little less in detail, independently a few months later.) The fourth was the first seminal paper on relativity. "Einstein was not yet done with 1905." He sent off a three-page fifth paper in late September. It showed that emitted light carries away mass from any source, and by 1907 he had it succinct and clear: $E = mc^{2}$. "Imagine the audacity of this step: every clod of earth, every feather, every speck of dust becoming a prodigious reservoir of entrapped energy."

Those with some taste for the x-language may find this book here and there at too simplified a physical-mathematical level, but the general reader can gain from it an honest "indication of the man" both by external event and by inward idea. We owe him that kind of attention. Beyond this volume the prepared seeker for Einstein can go next to the letters with Max Born, a lifelong exchange that was published in 1971 (and reviewed here) and then to the autobiographical notes. Hoffmann and Dukas have cleared a secure starting place.

"But we are shirking what cannot be shirked: on 6 August 1945 an atomic bomb was exploded over Hiroshima. Einstein's secretary heard the news on the radio. When Einstein came down from his bedroom for afternoon tea, she told him. And he said, 'Oh weh,' which is a cry of despair whose depth is not conveyed by the translation 'Alas.' "

6.6. Review by: Roy Johnston.
Irish Times (Saturday, 20-21 April 1973).

Lay readers wishing to get a feel for the implications of the General Theory Of Relativity will be captivated by this book, 'which succeeds in conveying both the internal beauty and the historical context of the work. It also conveys a picture of the man as a human being, a scientist, an artist and a person who was politically aware in a positive manner.

In a brief review, one can only touch upon the work. Few remember that it was not for his Theory of Relativity that he was awarded the Nobel Prize (in 1921) but for his theoretical explanation of the photo-electric effect in terms of a 'quantum of radiation' of which the energy depended on the frequency (ie colour) of the light. In 1921, Relativity was still too controversial.

This particular prizewinning work was only one of a group of remarkable pieces of original thinking which emerged from the Patent Office in Bern round about the year 1905. (Einstein didn't make the grade as regards academic appointments.) The others included the equivalence of mass and energy and the Special Theory of Relativity, which got rid of the concepts of absolute space and time and provided a satisfactory explanation for the null result of the famous Michelson-Morley experiment which attempted to measure the rate of drift of the earth through the (then postulated) 'aether'.

An element of this 'Special Theory' was the 'Fitzgerald Contraction' proposed by Prof G F Fitzgerald of TCD in or about 1898; this however, had the status of an 'ad hoc hypothesis' to explain away an embarrassing result. Einstein developed a theory in which this fitted naturally, with few basic assumptions.

Not being at home in the Prussian military atmosphere, Einstein took out Swiss citizenship. This granted him a certain immunity from the effects of the 1914-18 war, which he sat out in Berlin, having by this time achieved academic eminence. Here he showed political principle. Max Planck and 92 others signed a jingoistic German manifesto of intellectuals. Einstein, in relative isolation with three others, signed a 'Manifesto to Europeans' calling for cooperation among the scholars of the warring nations for the sake of the future of Europe.

From the depths of darkest imperialist barbarism in 1916 Einstein emerges with the General Theory of Relativity, providing thereby an explanation of gravitational force in terms of the underlying geometrical structure of the universe, conceived in a four-dimensional space-time. The 'square law' of Newton comes out to be, in essence, the same as the squares of Pythagoras. It becomes evident why Newton's law is not a cube, or the power of 2.5, a fact which hitherto had been accepted on a pragmatic basis.

Across the boundaries of war-torn Europe the excitement of this discovery percolated: de Sitter in Holland by 1917 was already experimenting with relativistic cosmologies; Eddington in England was able to drum up support by 1919 for an astronomical expedition to observe an eclipse and to verify one of the crucial predictions of the theory: the bending of starlight by passage through the gravitational field of the sun.

The way in which post-war Europe reacted to these discoveries, almost with hysteria, resulted in massive popularisation and world acclaim for Einstein. This stood him in good stead in the black thirties and forties, when he was able to use his influence help get many other Jewish intellectuals out of Germany.

To the end of his days he sought to extend the General Theory of Relativity satisfactorily to include the electromagnetic forces. He found many promising leads, but the theory never jelled satisfactorily. As for the quantum-mechanical world, which he had had a hand in setting up with his 1905 photo-electric work, and subsequently consolidating in the 1920s, with Bose (the 'Bose-Einstein Statistics'), he distrusted it profoundly. He argued for classical causality against the 'complementary principle' of Bohr and the Copenhagen School. (This argument was replayed by our J L Synge at the Irish physicists conference in Galway on April 8th, where the GOM referred to correspondence with Max Born and train-conversations with Compton; he unrepentantly struck to a pure Einstein position. There are profound implications in this for the philosophy of science).

Einstein in 1940 was one of the handful who realised the significance of the Hahn-Strassman experiments on uranium fusion, in 1939. He was instrumental, though a pacifist, in drawing to Roosevelt's attention the danger that the Nazis might make an atom bomb, thereby making the allied nuclear effort inevitable. He was appalled by Hiroshima and with Bertrand Russell in 1955 issued his last political statement to mankind - renounce war or destroy the race. Previous to this, he had sheltered by his influence progressive intellectuals who were persecuted by the McCarthy investigations. Despite pressure, and despite a strong emotional link with the Jewish refugees from Nazism, he avoided involvement with the infant State of Israel, although he had been associated with Weizmann in the twenties in fund-raising for the Jewish National Fund.

Einstein will forever rank among the giants as regards human intellectual achievement. Banesh Hoffmann's book will, I hope, convey something of this to many lay readers, thus helping to reduce the probability that in a new Dark Age his books will be ritually burned, as they were in 1933 in Germany.

6.7. Review by: Stanley Goldberg.
Science, New Series 180 (4086) (1973), 620-623.

The biographical sketch of Einstein by Banesh Hoffmann, may well go down as a classic, not only with regard to its characterisation of Einstein but also with regard to its technique of popularisation of science.
...
The most crucial element in this kind of popularisation, it seems to me, is to identify clearly the audience and, having done so, to devise a promising strategy and stick to it. Not everyone will be pleased, of course, but that's far better than leaving everyone displeased.

Hoffmann has obviously identified his audience as containing a sizeable proportion of people who know almost nothing about science or mathematics. Fortunately he does not make the error Louis de Broglie made in writing his Revolution in Physics. In order to avoid using mathematical symbols, de Broglie wrote out the equations in words. The effect of that strategy was not to eliminate the mathematics but rather to eliminate one of its major virtues, its conciseness. In Hoffmann's book, the mathematics is all but eliminated. Unless I missed some, there is only one equation in the whole book (Minkowski's fundamental invariant).

To anyone who knows a little science, Hoffmann's technique may at first be annoying. He tells his readers not to worry if they don't understand the descriptions of scientific theories in the book but to plough on unmindful of the subtleties. Hoffmann's strategy is clear. He has set out to convey the form and flavour of Einstein's work and to show how his work compared in these respects to that of his contemporaries and predecessors. In so doing he has often used a very large brush. He makes free use of analogies and drawings. Sometimes his analogies are outlandish (for example, the use of the ratchet device in a window shade roller to represent discrete energy levels in an atom), but usually they are not. Almost always they seem to me to accomplish what was intended, namely, to identify Einstein's philosophical position and personal characteristics and to relate them to his scientific contributions.

Sometimes the unsuspecting reader is sucked in in a most charming manner. Perhaps only Banesh Hoffmann would have had the nerve to introduce the concept of entropy with the following words: "When expressed more technically, the second law of thermodynamics involves a key concept, entropy, whose meaning fortunately need not concern us." Sometimes the issue is explained before the reader is told not to worry about it. Thus after a careful explication of Newton's concept of absolute time, Hoffmann asks rhetorically, "How then could the flow of absolute time be other than uniform if there is only itself as a standard against which to compare its flow?" and answers, "Never mind. The foundations of science are always a morass." Hoffmann goes on to note that Newton was no simpleton. He knew what he was doing and at the time it was bold and courageous.

There are times when Hoffmann uses his strategy of popularisation itself to lend power to his story. Thus in the midst of explicating Einstein's second paper of 1905, on determining the sizes of molecules, he breaks off with a remark that he realises that the discussion has been inadequate but we must move on "if we are to keep up with the head-long pace of Einstein's discoveries. ..."

It should not be concluded from this account that Hoffmann has failed to deal fairly with Einstein's scientific contributions. Sometimes his explications are first-class. This is the case with his explication of general relativity. His characterisation of the Einstein-Podolsky-Rosen paradox and Bohr's reply is brilliant.
...
There were at least seven biographies of Einstein published during his lifetime. Each had serious shortcomings. Einstein was against the publishing of biographies of living persons. He felt strongly that one's personal life should remain personal. These attitudes were reflected in the two biographies he not only tolerated but more or less sanctioned - Philipp Frank's and Carl Seelig's. After Einstein's death the publication of the Clark biography only revealed that his fears had been well founded. That biography was nothing short of a disaster. In treating Einstein's personal life with some distance, with grace, and in the context of his scientific work, Hoffmann has shown that it is possible to be enlightening without being gratuitously damaging.

6.8. Review by: Philip Peak.
The Mathematics Teacher 66 (3) (1973), 250.

It is important for teachers to present mathematics as a part of the world affairs and the people in it. As a biography of Albert Einstein, this book does just that. The author has clearly shown how Einstein used his intuitive powers, his feelings for theories, and his concentration on the problem until both feeling and intuition were satisfied. His thinking process is a good illustration that logic is only a part of the thinking process. The text brings out the importance of mathematics in developing the theories of science. Naturally, the book grows around the theory of relativity, but it also is an excellent discourse on how men think together, how governments are sometimes a hindrance to positive thinking, and how learned men sometimes fail to act in ways that reflect their learning. It is books like this that enable the reader to appreciate the impact of his discipline on the total society.

7.1. From the Publisher.

Modesty, humour, compassion, and wisdom are the traits most evident in these personal papers, most of them never before published, from the Einstein archives. The illustrious physicist wrote as thoughtfully to an Ohio fifth-grader, distressed by her discovery that scientists classify humans as animals, as to a Colorado banker, who asked whether he believed in a personal God. Witty rhymes, and exchange about fine music with Queen Elizabeth of Belgium, and expressions of his devotion to Zionism are but some of the highlights found in this rare, warm enriching book.

7.2. From the Foreword.

Albert Einstein was not only the greatest A scientist of his time but also by far the most famous. Moreover, he answered letters. And it is this combination that makes the present book possible.

Unlike our previous book, Albert Einstein: Creator and Rebel, this one is not a biography and does not explain Einstein's ideas. It has no chapters, no table of contents, no index, and, at first glance, no plan or structure. It consists, for the most part, of quotations from hitherto unpublished letters and the like that Einstein wrote without thought of publication. There is no need to describe them further here since they speak eloquently for themselves.

Some of the items were sent out in impeccable English, and these we have quoted verbatim. Other items were issued in less idiomatic English, and in presenting them we have made occasional minor changes while preserving the Germanic flavour that gives them charm. All other items are presented in English translation. Often an item that was issued in English was based on a German draft that still exists, and in such cases we have given the English version that was actually sent instead of making an independent translation.

Einstein was an artist not only in his science, which had a transcendent beauty, but also in his use of words. In the latter part of this book, therefore, we have included the original German versions or German drafts, whenever available, so that the reader acquainted with the language can savour Einstein's prose at first hand.

The quest for peace was an important part of Einstein's life. Indeed, a whole book, Einstein on Peace has been devoted to the subject, and so thorough is its coverage that hardly a scrap of unpublished material on the topic was left over for us to quote. Therefore, for details of this facet of Einstein we refer the reader to that book. We have, however, quoted a lengthy item from the book. Its inclusion has a twofold justification: it is a powerful statement in its own right with a special publication status: and its presence has a symbolic significance as a token salute to all the other items in Einstein on Peace that we were sorely tempted to republish here.

The order of presentation of the items is not haphazard. It is akin to that of crowding recollections of a rich life, each sequence apt to take unexpected turns as memory, with a logic all its own, leaps from remembrance to linked remembrance back and forth over the years. In this book there are several such sequences, their starts usually indicated by a more pronounced gap than usual between items. Each item may be taken by itself. But the book is intended to be read as a whole: it offers a seemingly rambling sightseeing journey whose cumulative effect, we hope, will be a deeper and richer understanding of Einstein the man.

For those who would like a road map, we have included a brief Einstein chronology at the end of the book.

7.3. Review by: Albert E Moyer.
Isis 72 (1) (1981), 147-148.

In presenting a selection of excerpts from Einstein's nontechnical and mainly unpublished writings, Helen Dukas and Banesh Hoffmann have abandoned literary conventions such as a table of contents, chapters, an index, and a methodical arrangement of topics. The result is a montage of Einstein's thoughts, designed to provide in its entirety an intimate and unconstrained impression of Einstein's character and personality.

The separate excerpts that compose this montage convey Einstein's opinions on a diversity of subjects: everything from Bach, quantum theory, and God to dialectical materialism, the death penalty, and academic competition - "an awful slavery no less evil than the passion for money or power." In tone the passages range from the humorous and optimistic to the cynical, outraged, and pessimistic "Man grows cold faster than the planet he inhabits." The excerpts, selected mainly from letters, show Einstein sharing his views with a variety of people. Correspondents include strangers with everyday concerns (a Jewish youth seeking advice on marrying a Baptist girl) as well as friends with international responsibilities (Queen Elizabeth of Belgium).

Dukas and Hoffmann preface each Einstein quotation with a brief commentary, at times interjecting their own recollections of their former associate. Limiting themselves mainly to explanatory details, they only occasionally offer broader historical appraisals. They echo, for example, the interpretation of Einstein as artist given in their prior collaborative work, Albert Einstein: Creator and Rebel (1972).

Excluding the final fourth of the volume, which contains the German originals of the English translations, one can read this short, informal book in an evening. Historians of science will want to spend such an evening, if only to get a hint of the riches at the Einstein Archives. Non-professionals and students will want to read the book to gain an introductory insight into the richness of Einstein's mind and personality.

7.4. Review by: Anon.
Change 11 (6) (1979), 60-61.

This short, lovely volume deserves a high place among the recent spate of books on Einstein. It reveals a scientist who answered his mail and did so with wit, warmth, and wonderful good humour. Here, from letters never intended for publication, we have Einstein on everything from the academic rat race for promotion ("I am not involved, thank God, and no longer need to take part in the competition of the big brains. Participating in it has always seemed to me an awful type of slavery no less evil than the passion for money or power") to when the world will end ("Wait and see!"). His fame brought letters from ordinary and well-known people the world over asking his views on religion, music, science, etc. An 18-year-old asks the purpose of man on earth and Einstein replies: "The answer is, in my opinion: satisfaction of the desires and needs of all, as far as this can be achieved, and achievement of harmony and beauty in the human relationships." A magazine sends him a questionnaire about Johann Sebastian Bach and he responds: "This is what I have to say about Bach's life work: listen, play, love, revere - and keep your mouth shut." Helen Dukas and Banesh Hoffmann, Einstein's secretary and collaborator, respectively, have done a splendid job of selecting and arranging these literary odds and ends. They even include this 1936 cornerstone message: "Dear Posterity: If you have not become more just, more peaceful, and generally more rational than we are (or were) - why, then, the devil take you."

7.5. Review by: Philip Morrison.
Scientific American 240 (6) (1979), 52-56.

Albert Einstein, the Human Side presents itself in such a modest and loving tone that it is fitting for the memory of the man it lets us hear. It is a fresh and delicious little anthology of citations from the body of Einstein's letters, journal entries and other written comment that is held in Princeton. Most of them are published here for the first time. Here is one, a "pious wish" written by the patient and generous physicist to store in a library cornerstone set down by an American publisher in 1936: "Dear Posterity," Einstein writes wryly, "If you have not become more just, more peaceful, and generally more rational than we are (or were) - why then, the Devil take you." These varied, penetrating, warm and open remarks to queens and schoolchildren, friends and antagonists, philosophers and sophomores have been sensitively chosen by two old friends of Einstein's and well translated. The German originals are included.

8.1. From the Publisher.

Using simple examples from everyday life, an Einstein scholar offers entertaining, nontechnical demonstrations of the meaning of relativity theory. Starting with the geometrical and cosmological ideas of the ancient Greeks, he traces the theory's development from its basis in work by Kepler, Galileo, Newton, Faraday, Maxwell, and others.

8.2. From the introduction.

Although this question will seem silly, consider it anyway: Why do the flight attendants on an airplane not serve meals when the air is turbulent but wait until the turbulence has passed?

The reason is obvious. If you tried to drink a cup of coffee during turbulent flight, you would probably spill it all over the place.

The question may well seem utterly inane. But even so, let us not be satisfied with only a partial answer. The question has a second part: Why is it all right for the flight attendants to serve meals when the turbulence has passed?

Again the reason is obvious. When the plane is in smooth flight, we can eat and drink in it as easily as we could if it were at rest on the ground.

Yes, indeed! And that is a most remarkable fact of experience. Think of it. In smooth flight a plane can be going at 1000 kilometres (about 600 miles) per hour relative to the ground, and yet inside the plane we do not notice any effect of this uniform velocity.

The same would hold if the plane were any other closed vehicle: There would be no interior effect of its uniform motion. This general statement is of central importance to our story. It is called the principle of relativity, and, as will be seen, it has had a strange history. At this stage there is little, if any, indication of how this principle could be related to nuclear energy, or to the possibility of a man being years older than his twin sister, or to the resolving of a discrepancy between the calculated and observed motions of the planet Mercury. Nor is there any indication of the changes the principle has led to in scientific ideas of time and space. But, as will be seen, the principle lies at the heart of a two-fold revolution that stands out as one of the glories of the scientific era.

Do not underrate the importance of time and space. They may seem to be intangible nothings, less palpable even than the faintest breeze. But they are the very stuff of existence; just try to imagine the world without them - or any world at all.

Shakespeare well understood the power and poignancy of time and space. Here, for example, is the song he wrote for Feste the clown in Twelfth Night (Act II, Scene III):

O mistress mine, where are you roaming?
O, stay and hear; your true love's coming,
That can sing both high and low;
Trip no further, pretty sweeting;
Journeys end in lovers meeting,
Every wise man's son doth know.

What is love? 'tis not hereafter;
Present mirth hath present laughter;
What's to come is still unsure;
In delay there lies no plenty;
Then come kiss me, sweet and twenty,
Youth's a stuff will not endure.

At first this seems artless and light-hearted, as befits the song of a clown. But read it again and note how Shakespeare devotes the first six lines to absence and reunion and thus to imperious space; note also how vividly he devotes the remaining six lines to inexorable time.

The physicist, too, is concerned with space and time. In his work he does not cry out "Journeys end in lovers meeting" or "Youth's a stuff will not endure." Instead, he talks of motion and rest, of distance and duration, and of centimetres and seconds. He measures intangible space and time, and by fitting them quantitatively into equations he makes them sport the formal mathematical garb of his craft. Yet the scientist is no cold automaton. Like the poet, he cannot create without emotion. Behind his equations are audacious imaginings and imperious feelings that transcend logic and give his science an artistry all its own - an artistry that can be made manifest without recourse to detailed mathematics.

Space and time are immensely powerful. In running from physical danger do we not confess the power of space by hoping to use mere distance space as a shield? Certainly if space can thus protect us, it is not a soft nothingness.

Nor is time. How safe we would be from death by nuclear bomb had we been born in the time of Shakespeare.

So familiar are time and space that we are apt to take them for granted, forgetting that ideas of time and space are part of the shaky foundation on which is balanced the whole intricate and beautiful structure of scientific theory and philosophical thought. To tamper with those ideas is to send a shudder from one end of the structure to the other. And to effect a profound change in them is to create a major revolution in science and philosophy. In his theories of relativity, Einstein revolutionised our ideas of time and space not once but twice.

Since relativity has roots that reach back to antiquity, Einstein does not officially enter our story until the penultimate chapter. Nevertheless, his ideas haunt these pages. Our tale is that of the historical path to relativity, with its open roads and seeming detours, and Einstein's ideas have been important in determining the attitude here taken toward the past.

In first reading what follows, focus on the overall picture. Details can always be returned to later.

Have a good trip

8.3. Review by: Robert March.
Physics Today (May 1964), 86-89.

The greatest strength of this short and very readable book, written by an award-winning Einstein biographer, lies in its thorough treatment of the "roots," the history of ideas about the relativity of motion. Though its tone suits a lay audience, even a working physicist may find some surprises in this narrative.

Relativity, Banesh Hoffmann reminds us, is as old as Western philosophy itself. Most of us know of Aristarchus of Samos, but he begins from Philolaus, whose moving-Earth (but not heliocentric!) cosmology came two centuries earlier. He touches briefly on the contributions of Nicolaus Copernicus, Johannes Kepler, Galileo Galilei and René Descartes on the way to the most "interesting" case, that of Isaac Newton. Here he dwells on the curious fact that the very thinker who enshrined relativity at the core of physical thought had grave misgivings about the principle and desperately defended his "absolute" space in the face of the very cogent criticisms of George Berkeley and Gottfried Wilhelm von Leibnitz.

Hoffmann gives Immanuel Kant and Ernst Mach their due but realises that it was the development of modern optics that posed the greatest challenge to relativity. On this part of the story he lavishes his most careful attention.

The problem arose from James Bradley's discovery, in 1728, of stellar aberration. In its time, it posed no conceptual difficulties, and in fact led to the best available estimate for the speed of light. But a century later, in the context of the wave theory of light and the ether hypothesis, stellar aberration pointed to a "stationary" medium and spawned the ill-fated attempts to measure the absolute velocity of the Earth optically.

François Arago made the first of these tries. In 1818 he sought the first-order effect on the index of refraction of glass as it turned in the ether "wind." It was the failure to observe this effect that gave rise to Augustin Fresnel's hypothesis of a "partial drag" of the ether, dependent on the index of refraction. This notion seems hopelessly contrived now, and it did even then. But after it was apparently "confirmed" by Armand Fizeau's ingenious moving-stream experiment of 1851, what was a natural philosopher to believe?

We must remember that in that era the wave theory of light, though widely accepted, had not yet silenced all of the sceptics. In the heat of this debate, it was hard for most scientists to regard light waves and the ether as separable hypotheses, regardless of how awkward the latter might sometimes appear. That is why Albert Einstein's more graceful analysis of the Fizeau result would become a strong selling point for special relativity.

When he deals with James Clerk Maxwell's mechanical ether, with its vortices and idle-wheels, Hoffmann's otherwise faithful historical account falters somewhat. It is not quite correct to represent Maxwell as discarding the model once it had served its purpose of guiding him to his equations. Physicists who would like to get this tangled story straight would do well to choose Daniel Siegel as their guide (in Conceptions of the Aether, edited by G N Cantor and M J S Hodge).

Like most good popularisers of Einstein's theories, when he comes to special and general relativity Hoffmann seeks a balance between formal rigour and intuitive clarity. In his initial discussions of each, he succeeds. For example, he explicitly shuns the common trap of confusing what is actually seen with the picture reconstructed after correcting for time-lags of light signals. But his transitions to formalism are sometimes a bit abrupt.

He resolves the "twin paradox" with the aid of Minkowski diagrams, which may satisfy physicists but have little appeal to others. And his introduction to the metric tensor will probably lose most lay readers. The text is too brief to do much more than display the bare bones of the formalisms, without sufficient applications to demonstrate their power.

When we strive to convey the meaning of relativity to the general public, physicists must never forget that however natural it may seem to us to regard the world as "a function of x, y, z and t," it rarely seems so to them.

Relativity and its Roots could be of some value as a supplementary text in an undergraduate relativity course or to anyone interested in the historical roots of contemporary science.

8.4. Review by: R W John.
Astronomische Nachrichten 306 (4) (1985), 250.

The special and general theories of relativity, both developed in the first quarter of the 20th century and integral to modern physics, revolutionised our understanding of space and time. In this book, the renowned scientist Banesh Hoffmann, also known as the co-author of books on Albert Einstein, whose assistant he was for a time at Princeton, illuminates the historical path to these theories. He describes the main currents and events in the development of astronomy, physics, and mathematics that, in their interrelation, ultimately led to the emergence of these theories, and explains their fundamental principles and elementary consequences. Naturally, looking back and in light of Einstein's ideas, the principle of relativity and the concepts of space and time are the primary focus.

The book is divided into six chapters, concluding with a name and subject index. In the brief introductory chapter, the author demonstrates how we encounter the principle of relativity in our everyday experiences and underscores its significance for our understanding of space and time. The following chapter takes us back to antiquity, presenting the astronomical world-views of Greek thinkers up to Ptolemy. The author then moves on to the Copernican system and the paradigm shift it ushered in, and arrives at Kepler's three laws of planetary motion. Galileo's astronomical discoveries and writings follow, as does, in the next chapter, his contributions to mechanics, particularly the discovery of the law of inertia. Finally, Newton's achievements are discussed: Newton's laws of mechanics and his theory of universal gravitation.

With regard to the concepts of absolute space and absolute time, with which Newton formulated his laws on a cosmic scale, the author aptly remarks on page 36: "They were dazzlingly effective inventions, and it was part of Newton's genius that, despite the problems associated with them, he dared to base his theory on them." As a principle of relativity in mechanics according to Galileo and Newton, it follows that reference frames moving uniformly relative to each other in a straight line are inertial frames - indistinguishable mechanically; the laws of classical mechanics are invariant under Galilean transformations.

In the following fourth chapter, the author, in accordance with historical development, addresses optics and electromagnetism. Römer's astronomical determination, that is, the simultaneous proof of the finiteness of the speed of light, was a fundamental empirical finding for the subsequent development of physics. Maxwell's theory of electromagnetism understands light as an electromagnetic phenomenon. A principle of relativity also applies to electromagnetic phenomena with respect to velocity, but here there is invariance with respect to the transformations of the reference frame named after Lorentz, which, except for the case of identical transformations differs from Galilean transformations. Does the principle of relativity therefore not apply to the connection between mechanical and electromagnetic processes? Optical and other attempts to determine an absolute velocity of bodies - the first relevant being the Michelson-Morley experiment - were unsuccessful.

Einstein then demanded that neither mechanical nor electromagnetic nor any other physical processes could distinguish between two reference frames moving uniformly in a straight line relative to each other, and, in conjunction with this, the independence of the speed of light from the motion of the source: This resulted in the special theory of relativity with its revolutionary consequences. It relativises the simultaneity of events and forces the abandonment of Newtonian concepts of absolute space and absolute time. The fifth chapter explores these and other consequences. According to the invariance of physical laws under Lorentz transformations required by the theory, classical Newtonian mechanics had to be modified, and relativistic mechanics took its place. Its quantitative results converge into classical mechanics for speeds small compared to that of light. Meanwhile, special relativity is an empirically well-confirmed theory that has also found its way into technology. What happened to Newton's theory of gravitation, which implies an instantaneous propagation of gravity? Special relativity does not include gravitation, but according to the theory, there should be no energy transfer at a speed greater than that of light (in a vacuum).

In the last chapter, we read about general relativity, Einstein's relativistic theory of gravitation. It is based on the principle of the general covariance of physical laws and the principle of local equivalence of inertia and gravity, which is required by the empirically established, highly accurate equivalence of inertia and gravity. The gravitational potential is then identified with the metric fundamental tensor of a Riemannian spacetime manifold. The existence of a gravitational field corresponds to a Riemannian spacetime for whose metric the Riemann-Christoffel curvature tensor does not vanish: gravitation is the curvature of spacetime.

B Hoffmann's book on the roots of both special and general relativity is a profound work, written by an experienced researcher with great expertise and attention to detail. The presentation of the sources used is meticulous and precise. Rich in facts and thought, it is densely written, yet at the same time transparent and engaging. The insightful analysis of the theories discussed and the clear elucidation of connections provide both excitement and stimulation. Numerous examples and instructive schematic drawings are included. All of this makes it a work that can be read with pleasure and profit by a broad scientifically interested audience.