## Reviews of Eric Temple Bell's books

**1. Algebraic Arithmetic (1927), by Eric Temple Bell.**

**1.1. Review by: Leonard Eugene Dickson.**

*Bull. Amer. Math. Soc.*

**34**(1928), 511-512.

This book of marked originality is of vital interest to advanced students in various branches of mathematics, including the theory of numbers, abstract algebra, elliptic and theta functions, Bernoullian numbers and functions, and the foundations of mathematics. A central feature is the new presentation of the author's principle of arithmetical paraphrases, which won him the Bôcher prize in 1924, jointly with Professor Lefschetz. ... A leading feature of the book seems to the reviewer to be its success in a systematic attempt to find a unified theory for each of various classes of related important problems in the theory of numbers, including its interrelations with algebra and analysis. An older case was the development of a theory of abstract groups which includes as special cases the really vital results and processes of the various theories of concrete groups, such as groups of permutations of letters, groups of linear or other substitutions, and groups of motions. The gain is not merely in great economy, but in the clarification which results from the exclusion of the irrelevant and the concentration on the relevant. ... This original and scholarly book is an honour to American mathematics.

**1.2. Review by: Herbert Westren Turnbull.**

*The Mathematical Gazette* **14** (194) (1928), 153-155.

This book deals with a thesis on the border lines of mathematics and logic, a region which has been extensively explored by the American school of mathematics during the last twenty years. Since the appearance of the great works of Whitehead and Russell on the principles of mathematics, which set many new trains of thought agoing on both sides of the Atlantic, there has been, particularly in America, a notable development of the basic principles of algebra. Among several experts, one of whom is the author of the book under review, the names of E H Moore and L E Dickson of Chicago may be mentioned for their special influence in this movement. ... Now some familiarity with the methods and terminology of Dickson's work is presupposed in the mind of the reader who takes up Algebraic Arithmetic, so that the book cannot form an introduction to this really fascinating subject without some extra help from without. Perhaps this is a little unfortunate, because in many respects the book is very suggestive. It is just the kind that is wanted in these days of specializing. It enables, let us say, the algebraist to learn something of the theory of number or of multiply periodic functions-and conversely. Also the author has broad views on the whole range of algebra; nor is he afraid of giving personal opinions, in delicately worded sentences, somewhat after the manner of Chasles or Klein, which throw a flood of light on the subject.

**1.3. Review by: Robert Daniel Carmichael.**

*Amer. Math. Monthly* **35** (7) (1928), 367-368.

Between the classic arithmetic on the one hand, as developed by the school of Gauss, and the modern analytic theory of numbers on the other hand, in which limiting operations usually appear both in the arguments and in the conclusions, lies an extensive intermediate region of the theory of numbers "where the methods of algebra and analysis are freely used to yield relations between integers expressed wholly in finite terms and without reference, in the final propositions, to the operations or concepts of limiting processes." To this part of the theory of numbers Bell has given the name algebraic arithmetic. It is not desirable that its boundaries should be sharply defined. In it may be placed many results in the theory of numbers which do not belong specifically either to the classical theory or to the modern analytic theory. During recent times, and especially in the hands of Bell himself, this part of the theory of numbers has had a rapid development. ... The book contains such a wealth of material and in such an abstract form that one can come to a full appreciation of its scope only by an extended study, supplemented by the development of many of its numerous applications. It is safe to predict that there will be a large development of algebraic arithmetic taking its rise from this book and from the researches of Bell on which it is based.

**2. Debunking Science (1930), by Eric Temple Bell.**

**2.1. Review by: Mark Hoyt Ingraham.**

*Amer. Math. Monthly*

**39**(3) (1932), 167-168.

Scientists, like other men, are constantly overestimating what they know. They repeatedly believe they have proved and disproved more than they have. When a scientific hypothesis works well in mentally organizing a few experimental results and fails only for experiments which are reported in small type or in obscure journals, it becomes a law of nature. A slightly changed cosmology, or a new biological discovery may lead to atheism; while the discovery of a very penetrating ray or an application of a non-definite quadratic form to physics seems ample justification for concluding that there is a God and that he is a pure mathematician. It is pleasant, therefore, to review even such a short booklet as this, warning scientists that their group self-respect should prevent them of all people from being overcredulous.

**3. The Queen of the Sciences (1931), by Eric Temple Bell.**

**3.1. Review by: T A A B.**

*The Mathematical Gazette*

**16**(221) (1932), 368.

The aim is, of course, to describe as simply as possible the main lines of mathematical progress during the last hundred years or so. The sections which tell how algebra was reduced to a set of postulates, so that variation of the postulates gives rise to numerous algebras, are particularly good, though to refer to "the British algebraic school, Peacock, Gregory, Sir William Rowan Hamilton, Augustus De Morgan, and others", is surely unfair to Boole. Most of us probably know those fits of depression in which we wonder if our teaching or research is really worth while; in such moods this book would prove an admirable tonic. We would also warmly recommend it to pupils just beginning to specialise; it will help them to broaden their mathematical understanding, and to develop a sense of the deep cultural significance of the science. Finally, we are indebted to Professor Bell for preserving that magnificent reply of Abel's when asked how he had been able to do so much in so short a time - "By studying the masters, not the pupils".

**3.2. Review by: Paul Henry Linehan.**

*Amer. Math. Monthly* **39** (5) (1932), 296-297.

Dr Bell, incidentally a successful novelist, has written in simple style, mathematically unconventional perhaps, but not flippant. ... The reviewer, frankly enthusiastic, would like to recommend the book to everybody. It would be an unimaginative person who would not, from even a cursory reading, catch some of the glorious spirit of mathematics. The teacher who has not enjoyed the advantage of extensive modern graduate study will be instructed and oriented. The apprentice in the guild of mathematicians will be inspired. The university lecturer will put the book down feeling that he has reaffirmed his faith.

**3.3. Review by: David Eugene Smith.**

*The Mathematics Teacher* **25** (4) (1932), 238.

It is a trite and rather patronizing statement that a certain book should be read by every high school teacher. If it were to be used at all, however, it might be used in connection with Professor Bell's charming little work, which came out a few months ago. It is a combination of historical information and material to show the nature of the various branches of elementary mathematics, with some excursions into such fields as complex numbers, trans formations, groups, algebraic numbers, transcendental numbers, and the in finite in mathematics. ... Professor Bell's skill as a novelist is carried over to enliven his writing upon somewhat abstruse subject. It would be difficult to find a better way of spending a dollar than that which leads to the buying of this book.

**4. Numerology (1933), by Eric Temple Bell.**

**4.1. Review by: Clarence Raymond Adams.**

*Amer. Math. Monthly*

**40**(6) (1933), 350-351.

[The author] writes, "Whoever believes mathematical theories of the universe to be anything more than convenient maps that may be radically revised or torn up at any moment is a numerologist. A scientific worker who holds no such belief is a scientist. There is of course no stigma on either term." Numerology would seem to bear a relation to mathematics much like that of astrology to astronomy or of alchemy to chemistry. ... 'Numerology' is written in a breezy and entertaining style, and no doubt will soon be offered as evidence that California produces, along with choice prunes and other things, bigger and better satires. Superficially the book is non-technical; it will be found amusing and informative by almost any layman with an interest in science. Most professional scientists, also, will find the book interesting; some few of them may be incensed, especially if they interpret the section en- titled A Request in Chapter VII as a not very subtle inference that among American scientists Professor Bell himself is alone free from numerological tendencies.

**5. The Search for Truth (1934), by Eric Temple Bell.**

**5.1. Review by: F Stuart Chapin.**

*Journal of the American Statistical Association*

**30**(191) (1935), 631-632.

This delightful and stimulating book is well worth the attention of all mathematical statisticians who are interested in the fundamental problems of method. It abounds in epigrammatic statements of such argumentative flavour as: "2440 A.D. will have forgotten everything about us except that we were a particularly moronic species of homo sap.,"; "Science seldom makes a grand attack on a whole army of difficulties; it divides and conquers,"; "Theory and experiment are the twin crutches by which science hobbles forward,"; "not homosexuality, but extrapolation, is the most characteristic vice in which our age has imitated classic Greece,"; "For all of three centuries the new masterpiece, ... (the second half of Aristotle) budded upon itself in a furious profusion of incredible gargoyles,"; and "He (Leonardo) was one of the first to see clearly that deductive reasoning alone will get nothing but meaningless noises out of nature". Bell traces the activities of "homo sap." in the evolution of deductive reasoning and the relation of this activity to what men mean when they use the word "truth." The book is an abbreviated intellectual history of western civilization in four phases as these are reflected in and symbolised by advances in fundamental logic and mathematics. Is there such a thing as truth? and What is the truth? were merely different phrasings of a question that has misled man for 6000 years. ... In spite of digressions, that are sometimes annoying, the book will repay more than one reading.

**5.2. Review by: H T C.**

*The Journal of Philosophy* **32** (5) (1935), 134-135.

The review of this book must be a personal reaction. I enjoy jokes, including some of Professor Bell's, but not when they are ill-natured or based on misinformation. The author is a mathematician of some distinction, and his book a sketch of certain aspects of the history of science. Incidentally, though wordily, it is a satirical attack on all philosophers. Philosophers engage in verbalization; whereas science is based on experiment and operational definitions. Plato is particularly bad, with his number-mysticism and absolute truth, hoary hiding places of traditional errors. This sort of thing I consider unintelligent, the essence of bigotry rather than of scientific caution. It is unintelligent, because the author is not able to discriminate his real friends from his enemies. ... I am not at all sure how Professor Bell is going to bring to the test of any experiment or observation his own favourites among contemporary theories, the transfinite numbers and the plurality of new logics. If his own criteria of meaning are sufficient, do they have any meaning?

**5.3. Review by: Oliver L Reiser.**

*Philosophy of Science* **2** (1) (1935), 118-120.

This book retraces the intellectual evolution of the human race in its struggle for a science of thinking. It reveals the pinnacles which have been reached during the last four thousand years and exhibits what human beings in successive eras have imagined truth to be. But the end is not yet! For the ancient query of Pilate, What is Truth? is still unanswered. New answers are yet to be considered. ... Bell's book is stimulating, iconoclastic and devastating. Gilded with literary iridescence and sparkling with original insights, the book takes the new logic seriously and nonchalantly transcends any inherent contradictions. ... a word of warning to the author. He has put himself on the spot, so let him guard carefully his protoplasmic integrity! Bell hoists many heads upon the petard of ridicule. He derides philosophers ...

**5.4. Review by: Richard Bevan Braithwaite.**

*The Mathematical Gazette* **19** (234) (1935), 239.

The best I can do to describe the book in one sentence is to say that it is an attempt to communicate the present atmosphere and some of the relevant history of mathematical physics by a mathematician of an empirical turn of mind with a sceptical temper who uses magnificently violent language. Professor Bell detests speculations that cannot be tested at this moment by experiment-the Expanding Universe, for example, about which it is ridiculous to say anything one way or the other until Hubble's 200-inch mirror telescope is in action. Professor Bell's sceptical empiricism is surely not the whole truth about scientific method (for what is the experimenter to confirm or refute except a theory previously thought of?); but it is a salutary antidote to the popular works on science which treat of far-reaching cosmic speculations as ascertained facts.

**5.5. Review by: Leslie M Pape.**

*Annals of the American Academy of Political and Social Science* **186** (The Attainment and Maintenance of World Peace) (1936), 237.

Bell rightly disparages Pilate's undeservedly famous query, "What is truth?" The question means for him a series of ad hoc questions of the form, What results will follow from such and such an operation or set of operations? Of course, the question might also mean, What is absolute truth, or the secret of the universe? But this is next to nonsense. Again, it might mean, What is the criterion of truth? Finally, it might even mean - and this is what it should mean first of all - What is the meaning of the word "truth"? ... Despite its popular, not to say tipsy, style, the book is an excellent analysis and history of important mathematical and logical concepts.

**6. The Handmaiden of the Sciences (1937), by Eric Temple Bell.**

**6.1. Review by: A M Ginsburg.**

*The Mathematics Teacher*

**30**(3) (1937), 141.

Considering the plethora of excellent popular accounts of recent developments in the various sciences, it is high time we had something of the kind in mathematical physics. And that is what we find in "The Handmaiden of the Sciences." This book is intended as a companion volume to "The Queen of the Sciences," written in 1931 and now unfortunately out of print. Whereas the older book provides us with an airplane ride over the territory of modern pure mathematics the author now takes us on a similar excursion over the field of applied mathematics. The title is perhaps a slight misnomer since the applications here discussed are almost entirely in the domain of physics. Lucidity of exposition, liveliness, a certain humanness of treatment, and an infectious élan are the outstanding qualities of this book. ... Teachers of mathematics eager for a widening of horizons will find it well worth their attention.

**6.2. Review by: George Abram Miller.**

*National Mathematics Magazine* **12** (2) (1937), 102-103.

This is a little book on a very big subject and treats briefly a large number of mathematical topics. It presupposes very little technical knowledge and the reader should not expect to secure much such knowledge from it in view of the small amount of space devoted to each subject. Its object seems to be to inspire the reader, but it fails to provide much assistance to the reader who might become interested in securing additional information along various lines. It contains a brief table of contents but no index. Fortunately its author is a man of high mathematical standing and gives his views with great clarity even if they may sometimes be biased.

**6.3. Review by: Frederick Bernard Pidduck.**

*The Mathematical Gazette* **22** (248) (1938), 92-93.

This book aims at describing, for the general reader, the service which mathematics has rendered the sciences. One would know better how to review it if one knew who would read it; and we are all aware that the world is obtuse to the successes of mathematics and ignorant of its nature. So that when the book falls into the hands of the right reader, he will find in it much to stimulate him and much good information on the aims of mathematics. It was a good idea, for example, to put in boundary conditions and an introduction to invariants. And there are many passages in which the author writes clearly and boldly and lays out the nature of mathematics in a way that can hardly fail to grip the semi-scientific reader, or the ordinary reader of intelligence. The author is well aware of the futility of some methods of teaching, and of the uncertainty and lack of exactness of mathematical reasoning; well aware also of the feelings of the researcher. In some ways the book will leave too much the impression of a person who has approached the modern physical theories from the mathematical point of view. There is too much logic, too much stress on the logical basis of mathematics and too much on those parts of mathematics which depend on logical assumptions; too much also about the Greeks and their way of thinking. ... Sometimes we have passages of real eloquence, followed at once by atrocious jokes and colloquialisms; and the worst of it is that the jokes are not always appropriate but apt to be dragged in after the manner of an after-dinner speaker or newspaper hack.

**6.4. Review by: Samuel Eugene Rasor.**

*Educational Research Bulletin* **18** (1) (1939), 20-21.

Mr Bell is the author of many books, some of which are fiction. While his style is indeed different - flashing, brilliant, humorous, even flippant - it is never dull. ... The author attempts, by illuminating examples and general principles, the suggestion that mathematics is indeed the efficient servant of the sciences. ... Here in this book one finds high lights that are dramatic with illuminating penetrations, all in a few hundred pages of fascinating reading such as is seldom encountered. ... There are enough challenges and thrills encountered here to last the reader through a great many interesting and dramatic sessions of intellectual pastime.

**7. Men of Mathematics (1937), by Eric Temple Bell.**

**7.1. Review by: George Sarton.**

*Isis*

**28**(2) (1938), 510-513.

There are many ways of dealing with such an immense and many-sided subject as the history of mathematics, and whichever one's point of view, one might try to write a consecutive and relatively complete account or restrict oneself to selected parts. The author of this book has chosen the second alternative, and has simplified his task considerably by a double restriction. He has selected some men of mathematics, and for each of these he has selected some achievements. The first selection is somewhat arbitrary, the second, unavoidable if one aims at satisfying as large a potential audience as possible. There can be no quarrel about that, though one cannot help wishing that a writer endowed with as much mathematical insight and literary talent as the author would address himself to the more difficult task of writing a history of mathematics covering, however briefly, the whole field. Such a book is badly needed. In the meanwhile the present one will render great services and the more so because it is very largely devoted to the last century and a half, a period generally neglected in the historical textbooks. ... The work is very readable, but it would have been even more readable, if the author had not tried to enliven his style with cheap jokes and silly remarks. ... As the author does not quote his sources it is difficult to measure his accuracy, and his book can be used only for reading, not for reference or study. The most valuable parts of it are not the purely historical, the correctness of which cannot be readily ascertained, but the mathematical remarks, drawn from his own rich experience. ... The author had it in him to write a book of permanent value; he preferred to write one that would sell right away.

**7.2. Review by: Edwin Bidwell Wilson.**

*Science, New Series* **87** (2269) (1938), 578-579.

If one thumbs the numerous cards for Eric Temple Bell in the Harvard College Library, one finds intermingled with those representing his mathematical work some endorsed "John Taine, pseud." and representing something else - "thrillers." Thus does a great library override the author's modest pseudonymity. The Jekyll-Hyde characteristics of the Bell-Taine contributions are both present in Bell's "Men of Mathematics," but what part of the work will be attributed to Jekyll and what to Hyde will vary with the reader. For that large number of the somewhat general public to which Simon and Schuster cater in some of their publications it will be Dr Jekyll who writes the "heart-interest" material and the glittering generalities in an often loose style and it will be Mr Hyde who tries to expound the theory of algebraic ideals or of transfinite numbers or of symbolic logic, whereas for the professional mathematician the attributions for these respective parts will be inverted. ... There is no doubt Bell's work is readable, interesting and generally good; what it needs is some kind friend who will draw a firm blue pencil through an adjective here, a phrase there and occasionally a paragraph to the end that the work might attain that sort of precision of statement which would be in the true spirit of mathematics and to the further end that Eric Temple Bell, member of the section of mathematics of the National Academy of Sciences, might be protected from some of the cheaper vagaries of John Taine. It would be a mistake to assume that the popularity of the book need suffer thereby.

**7.3. Review by: T A A B.**

*The Mathematical Gazette* **21** (245) (1937), 311-312.

Every school library, every young mathematical enthusiast, every teacher of mathematics ought to get this book. I cannot imagine that anyone who opens it will fail to read any word of it. There are numerous errors of detail, unimportant in a stimulant; there is a trace, it seems to me, of an error in principle; but the author succeeds in his main aim, to portray mathematicians three-dimensionally by setting them in or against their historical background, social, cultural and philosophical. ... To all, mathematicians or not, who wish to see these great names as men, I can recommend this volume with full confidence that every page of it will be found interesting and informative.

**7.4. Review by: Guy Waldo Dunnington.**

*National Mathematics Magazine* **11** (8) (1937), 406-407.

There exists today a strong tendency to pay homage and attention to the heroes of science, to humanize its essential creators, and this book is good evidence of the fact. ... the aim of the book is popularization, or an appeal to the general reader. He specifically states that it is not intended to be a history of mathematics, or any section of it. ... Dr Bell is a seasoned, skilful writer with a fluent style; he writes with a realistic, curt, potent wit and stark, frank humour which does not stop short of vigorous, rollicking slang (regarded by some as "undignified" - but this is a matter of taste). Doubtless this will amuse many casual, rapid readers, but it also leads him to a certain type of exaggeration. ... Dr Bell allows his imagination to play; conjecture, personal opinion, and speculation are abundant throughout these chapters as to what the historical development would have been if conditions had been different, or some mathematicians had lived longer, behaved differently, or if mortals were constituted otherwise.

**7.5. Review by: Louis Weisner.**

*Science & Society* **1** (4) (1937), 579-580.

In this book a creative mathematician undertakes the task of portraying to the general reader the lives of the great mathematicians from Zeno to Poincaré, with the object of describing the sort of human beings that mathematicians are. The author clearly demonstrates that creative mathematical ability may be accompanied by almost any combination of virtues or vices; that great mathematicians may be great or mean in other fields of human endeavour; that the lives of mathematicians may be serene and tranquil or sorrowful and tragic; and that "on the whole the great mathematicians have lived richer, more virile lives than those that fall to the lot of the ordinary hard-working mortal." But Professor Bell not only tells us the personal story of individual mathematicians and of their relations with their contemporaries, he expounds through his heroes the development of the fundamental concepts of mathematics, such as number, group, invariant, infinity and dimensionality. This achievement is as unique as it is important and is in direct contrast with the works of those historians who set out to describe the history of mathematics and end with short biographies of and anecdotes concerning mathematicians.

**7.6. Review by: Anon.**

*The Journal of Higher Education* **9** (1) (1938), 53.

It is indeed fortunate that the preparation of a book of this kind should have enlisted the interest of one of the foremost mathematical scholars in the land. No one but a person of both wide and deep scholarship could have written a volume which is not only an intensely human document, but at the same time the story of the steady march of mathematical ideas from the time of the Greeks to the present. ... It cannot be too strongly urged upon all teachers of mathematics, particularly those in collegiate work, that they read this book carefully and repeatedly. They will be far more intelligent and helpful teachers if they do. The narrow escape of some of the most original minds in the history of mathematics from complete frustration through the inability of teachers to recognize genius when they saw it ought to put every teacher on his guard against allowing rules and precedents to stand in the way of education.

**7.7 Review by: Lao Genevra Simons.**

*Amer. Math. Monthly* **45** (1) (1938), 43-44.

The reader of Men of Mathematics lays down the book after a first reading with a feeling of profound satisfaction that here is a fascinating set of short stories (it is more than a series of biographies), for frequent perusal in parts or as a whole, a book of reference for historical use, and a means of forming friend- ships with men already more or less well known. The work does not claim to be a history of mathematics but to show what "sort of human beings the men were who created modern mathematics," "the importance for modern mathematics of the man's work" and "the human appeal of the man's life and character." ... This book may be read with appreciation by intelligent laymen who are un- familiar with the field of mathematics. One hopes that it may serve to revise their notions of mathematicians ... Time and again, phrases are used whose aptness is extraordinary.

**7.8. Review by: H F M.**

*The High School Journal* **20** (6) (1937), 243-244.

'Men of Mathematics' presents to the reading public brief biographical sketches of three of the great ancient mathematicians, Zeno, Eudoxus, and Archimedes, and twenty-seven of those who made great contributions in this field from Descartes to the present time. ... Through the biographies of these men Dr. Bell has portrayed to the reader not only their lives but also the gradual growth and development of the science of mathematics. He has done this in such an interesting style, and has interspersed so much of humour withal that the whole is of intense interest to the general reader as well as to the mathematician. In other words there is so much of "human interest" in the book that it is a splendid introduction for the general public to the history and development of mathematics. The emphasis is on modern mathematics. The appeal is to anyone who wants to know of its development.

**7.9. Review by: W D R.**

*The Mathematics Teacher* **31** (2) (1938), 85.

This very interesting new book is not a history of mathematics in the strict sense although the story deals especially with those mathematicians who created the Golden Age since Newton. The main interest for the reader throughout the book lies in the personalities of the men described. 'Men of Mathematics' should be accessible not only to those with a special interest in mathematics but to the general reader who wishes to understand how the race has progressed through the years. It is strange that historians, for ex ample, have given so much attention to the war lords and their achievements in view of the damage they have done to civilization and have said so little about the great men of mathematics and science like Newton and Pasteur to whom we are greatly indebted. This book will make available a rich source of material for all those who wish to change the emphasis from men of war to men of peaceful pursuits.

**7.10. Review by: Alonzo Church.**

*The Journal of Symbolic Logic* **2** (2) (1937), 95.

After an introductory survey, and a chapter on Pythagoras, Zeno, Eudoxus, and Archimedes, the book contains a collection of brief biographies of great mathematicians from Descartes to Poincaré and Cantor. Of especial interest to logicians are the chapters on Leibniz, Boole, and Cantor. It is unfortunate that these very readable and well written chapters, which give on the whole a correct picture of the importance which foundational criticism and with it the method of symbolic logic have acquired in modern mathematics, are nevertheless in certain particulars inaccurate or misleading.

**8. Men of Mathematics (2 volume reprint) (1951), by Eric Temple Bell.**

**8.1. Review by: T A A B.**

*The Mathematical Gazette*

**37**(322) (1953), 320.

Bell's lively and stimulating collection of mathematical biographies should be in every school library. Penguin Books must be warmly thanked, not only for giving us this cheap edition at the price of an ounce of tobacco, but also for recognising that mathematics is neither the esoteric playground of an eccentric minority nor a collection of amusing puzzles. The schoolboy, the ordinary citizen, even the mathematician, who would learn something about the personalities of the subject, can not do better than begin with these volumes. Bell's facts are sound, his style is racy, his enthusiasm infectious, and his opinions, whether we accept them or not, are at any rate not insinuated in any half-hearted deprecating fashion but set down clearly and forcefully, often with a humour gently malicious. Informative, exuberant, provocative, ...

**9. The Development of Mathematics (1940), by Eric Temple Bell.**

**9.1. Review by: I Bernard Cohen.**

*Isis*

**33**(2) (1941), 291-293.

Mr Bell has long been determined to write a history of mathematics and his recent books may be represented as a series of successive approximations to a genuine history. Within one restriction, the present book is excellent - that restriction consists in the fact that it really begins on 99 with Ch. 7: "The beginnings of modern mathematics, 1637-1687." In the first 98 pages, there are many statements that one will take exception to. ... For the rest (that is, what I call the book proper) Mr Bell has given us the best concise account of the development of mathematics in any language. With a veritable genius of exposition he has rendered his subject matter clear and interesting. ... Mr Bell's book, within the restriction imposed above, is a valuable book, of interest to the professional mathematician, but most of all to the student of mathematics who may gain by its means a clear insight into his intellectual heritage. To the student this account is doubly worth while, for it not only gives him useful, entertaining, and enlightening accounts of the history of his subject, but it also satisfies the function outlined by Mr Bell in his lecture on "Fifty years of algebra in the United States, 1888-1938": "If there is anything that will give a man, young or old, a decent humility and a sane humour regarding his own efforts, it is an acquaintance with the work of his predecessors and contemporaries. The earlier this acquaintance is gained, the better for all concerned, including mathematics."

**9.2. Review by: T A A B.**

*The Mathematical Gazette* **25** (265) (1941), 198-199.

In recent years, Professor Bell has written a number of books about mathematics, suitable for those of us who would like to know more of the modern developments, even though we cannot hope to study them in the original memoirs of the current periodicals. This latest work may be regarded as complementary to 'Men of Mathematics' (1937) but it does not seem to me to be quite so good a book. There are two main reasons for this judgment. One is that clearly the author must assume that much of the mathematics he discusses is known to his readers; he has made a considerable and often successful struggle against this handicap. The second reason is that the many comments are often irrelevant and sometimes misleading. ... there is much sound information and much good entertainment to be had from this rather expensive but beautifully printed and neatly bound volume.

**9.3. Review by: Rudolph Ernest Langer.**

*Science, New Series* **93** (2412) (1941), 281-283.

"Once we venture beyond the rudiments," says Mr Bell, "we may agree that those who cultivate mathematics have more interesting things to say about it than those who merely venerate." No more eloquent substantiation of this assertion could be wished for than this book in which it appears. A cultivator himself, its author requires no introduction to mathematicians. He knows mathematical creation - its trials and its rewards - at first hand. Nor does he need introduction to the wider reading public. It seems to this reviewer, however, that in this work he has risen to a new level of accomplishment, which merits the genuine appreciation of all those who regard mathematics and its related sciences as a vital field of human activity, and find interest in the history of their development. This is an eminently readable book, written in an engaging and graceful style. At the same time it is a scholarly work with a wholly serious purpose, full of information and fact, and covering much material which is otherwise not easily accessible.

**9.4. Review by: Alonzo Church.**

*The Journal of Symbolic Logic* **5** (4) (1940), 152-153.

This is, in the author's words, "not a history of the traditional kind, but a narrative of the decisive epochs in the development of mathematics." "Only main trends of the past six thousand years are considered, and these are presented only through typical major episodes in each." ... Professor Bell has written an extremely valuable and stimulating book, whether considered from the point of view of mathematics generally or from that of logic and foundations in particular. There are, however, from the latter point of view a number of serious errors of detail.

**9.5. Review by: John M Reiner.**

*Philosophy of Science* **8** (3) (1941), 464-465.

Dr Bell has written a book, not on the history of mathematics, but on the anatomy and morphogenesis of mathematical thought, something of far more importance. The result will be of interest and value not only to students of mathematics, but to their professors as well, and it is heartily to be commended to scientists in all fields. ... Dr Bell has really opened up his subject and examined its organs and tissues, recorded their growth and function, and hazarded some shrewd guesses about the mechanisms at work in the process. It is worth recording that in the present work Dr Bell has exercised some control over his customary inveterate tendency to make what might vulgarly be termed learned wisecracks. The result has been to give added significance and pungency to those that remain - and they are many.

**9.6. Review by: M G K.**

*Journal of the Royal Statistical Society* **104** (2) (1941), 176-177.

The excuse for noticing this book in a statistical journal lies in Professor Bell's last chapter, but it would be unjust to pass over the previous 22 chapters in complete silence. This is a stimulating and, in places, a fascinating book to the mathematician. It covers the period of mathematical history from the earliest times, but is no ordinary chronicle of mathematical events and memorabilia. It is a history of ideas, not of men or of theorems, and as such is a notable advance on certain existing histories. Some of the judgments expressed are highly individual, but by no means arbitrary; and the author is remarkably up to date at both ends of his time-scale ....

**9.7. Review by: E N.**

*The Journal of Philosophy* **38** (5) (1941), 137-138.

This book has been written primarily for practicing mathematicians who are interested in learning something of the major trends and episodes in the development of their discipline and of the way in which current mathematical concepts and techniques have come into being. But students of philosophy and of the history of formal logic will be well repaid for consulting it. They will find in it not only an engrossing story of important stages in the construction of mathematical methods; they will also find a clear survey of the development of mathematical logic from its origins down to the present day and, what is perhaps even more valuable, an intelligible account of the influences which technical problems of mathematics have exercised upon the direction of logical inquiry. ... Professor Bell's appraisals and accounts of the work that has been done on the foundations of mathematics and logic are not uniformly sober or adequate. He enjoys being irreverent at the expense of stuffy men and philosophies, and he likes to provoke his readers by presenting them with what seem to him like unorthodox conclusions. This is good fun-up to the point when his enthusiasm for leg-pulling gets the better of clarity in formulation or balanced judgment.

**9.8. Review by: Guy Waldo Dunnington.**

*National Mathematics Magazine* **16** (8) (1942), 415-416.

This book is designed to be a broad survey and historical analysis of the growth of mathematics; certain philosophical elaborations are also present. It is not a history in the usual sense of the term, but does suggest new connections between old ideas or new applications of old methods. It may provide the reader with a deeper grasp of the entire field. There is a laudable attempt to present mathematics within the framework of general history, covering the period from circa 4000 B. C. to 1940.

**9.9. Review by: Burton Wadsworth Jones.**

*Amer. Math. Monthly* **48** (2) (1941), 140-141.

This book was written in response to a "request ... for a broad account of the general development of mathematics." 'Men of Mathematics' was a history of mathematics in terms of its men. This book is a history of the evolution of mathematical ideas; in the author's words, an "endeavour to portray mathematics as the constantly growing, human thing that it is, advancing in spite of its errors and partly because of them." ... Readers of books by this author are so numerous that it is scarcely necessary for the reviewer to comment on his refreshing style flavoured with the spice of many a debunking quip ... In the opinion of the reviewer, aside from its value as a pleasant-to-read history of the evolution of mathematics, this book is especially worth while in its careful implantation of a few very fundamental ideas about mathematics as a whole.

**9.10. Review by: Dirk Jan Struik.**

*Mathematical Reviews*, MR0002768 **(2,113b)**.

Bell's new book is a source book of information on a vast field of modern and past mathematical research. ... Authors and teachers working on a particular subject will do well to consult Bell's book in order to obtain a broad survey of its outstanding results and the main authors up to the present time. ... The experience of the author as a creative mathematician, a teacher and interested colleague has made it possible to place lively comments, pithy summaries and challenging outlooks between an otherwise factual survey of achievements. These surveys have unequal merits, excelling in arithmetic and algebra and in related fields, and losing their completeness somewhat in regions less professionally familiar to the author.

**10. The Development of Mathematics (2nd edition) (1945), by Eric Temple Bell.**

**10.1. Review by: Saunders MacLane.**

*Amer. Math. Monthly*

**53**(7) (1946), 389-390.

This magnificent, inclusive, and provocative survey of the origin and adventures of mathematical ideas has now appeared in a second edition. Various material has been added ... The great virtue of this book is that it does not merely record facts, but it arranges ideas and passes judgment as to their importance. This aim, combined with the tremendous scope of the work, makes it inevitable that there should be errors both of fact and of judgment. ... It's great fun to read this book, just because there are so many chances profitably to disagree with its provocative author. The wealth of possible topics of difference must be read to be appreciated. ... The book is of great value for many classes of readers. The specialist will find a new perspective and enjoyable provocation in his own field. The historian will find a writer with the courage to attempt a rational assessment of modern mathematics. The student will find a wealth of suggested new vistas. The teacher will find the historical origin and (possibly) the lack of significance in the material appearing in courses. The writer of textbooks will find new suggestions, well barbed (elementary books still fail to prove Taylor's theorem in the same way in which Taylor failed). The philosopher will disagree with the jabs at Kant, but will profit from the view of living mathematics. The young mathematician will gain background and will learn of the ebb and flow of fashion in the specialties of research. To all these and others one might say: don't wonder about it, but go, read, and disagree for yourself.

**10.2. Review by: Alonzo Church.**

*The Journal of Symbolic Logic* **12** (2) (1947), 61-62.

The new edition has many amendments and additions, designed either to remedy shortcomings of the first edition or to take account of new developments between 1940 and 1945. ... In a work of this magnitude, minor historical and mathematical errors are difficult to exclude. Not all those of the first edition have been corrected, and some others appear in the new material.

**10.3. Review by: Dirk Jan Struik.**

*Mathematical Reviews*, MR0016318 **(8,1b)**.

This edition differs from the first, which appeared in 1940, in that about fifty pages of new material have been added. The additions include numerous short amplifications of miscellaneous topics together with longer notes on subjects in which there have been striking recent advances, such as symbolism, algebraic and differential geometry and lattices.

**11. The Magic of Numbers (1946), by Eric Temple Bell.**

**11.1. Review by: Adolf Prag.**

*The Mathematical Gazette*

**31**(297) (1947), 304-305.

It reads like a racy historical novel, full of dramatic life-story detail, full of witty, sarcastic remarks and innuendos; it often reads merely like a (somewhat uncharitable) History of Human Errors - errors due to the timeless desire not only to know but also to understand. Historians are often admired for their ability to create the contemporary atmosphere in their description of a certain period of the past. Bell's technique is exactly opposite, he brings the atmosphere of his own environment into the past ... [The reader] must on occasion be content to have some historical claim substantiated by simple repetition; he should remember his Undergraduate Society debating the eternal "Hen-or-Egg" Priority problem; but before all else he must be mature enough not to be muddled, as so many of us are, by irony and sarcasm - and he must certainly not be pedantic about detail. He will like and enjoy the book greatly for its stimulating, provoking, exasperating style - he will be thankful to be urged to think for himself again, rather desperately at times, to keep, or to find, his own bearings in this timeless conflict of evidence and reasoning, intellect and insight, knowledge and vision, experiment and pure thought; laying aside, if possible, his own not quite so far Western idiosyncrasies.

**11.2. Review by: Duane Studley.**

*Mathematics Magazine* **21** (3) (1948), 146.

While considering Pythagoras' numerology Bell repeatedly asks the question were numbers discovered or invented? The book doesn't reach a definitive answer to the question but the case for discovery seems more plausible after the reader has ploughed through the story ...

**11.3. Review by: Eduard Jan Dijksterhuis.**

*Mathematical Reviews*, MR0018594 **(8,305a)**.

In this book two divergences of opinion which keep scientists divided in our days are traced back to their alleged origin in Greek thought. (1) Is mathematics to be regarded as a free human creation or as an exploration of a reality existing somehow outside us? (2) Is it really impossible to obtain a reliable knowledge of the physical world without observation and experiment or are we to believe Eddington's contention "that all the laws of nature that are classed as fundamental can be foreseen wholly from epistemological considerations"? The investigation leads back to Pythagoras and the development of his views in the philosophy of Plato. The opinions of both philosophers are elucidated and discussed; moreover their lives and adventures are related at length.

**12. The Magic of Numbers (Reprint) (1991), by Eric Temple Bell.**

**12.1. Review by: Craig B Merow.**

*The Mathematics Teacher*

**85**(6) (1992), 493.

For the past eighteen years I have regaled many a class with stories gathered from E T Bell's classic 'Men of Mathematics'. With the recent publication of a Dover edition of his equally engrossing 'The Magic of Numbers', my repertoire of mathematical tales has been significantly enlarged. Bell is a wonderful storyteller, and his subject, the Greek philosopher and mathematician Pythagoras, supplies him with ample material to demonstrate his skill. Pythagoras, working in the sixth century B.C., pioneered the use of experimental investigation to discover the laws of nature but later abandoned this approach in favour of an introspective, deductive, mathematical mysticism. Bell traces the history of these two paths to knowledge from ancient Greece to the twentieth century. Along the way he treats the reader to the personal stories of a host of outstanding mathematicians, philosophers, and scientists. ... this book is highly recommended for mathematics teachers with a philosophical and historical bent and for like minded students interested in improving their vocabulary.

**13. Mathematics: Queen and Servant of Science (1951), by Eric Temple Bell.**

**13.1. Review by: Ernest Nagel.**

*Science, New Series*

**113**(2938) (1951), 445-446.

Dr Bell's present book is an integrated and considerably enlarged version of his earlier 'The Queen of the Sciences' (1931) and 'The Handmaiden of the Sciences' (1937). No one who has ever seen his publications intended for general audiences need be reminded that he can write on difficult matters clearly, informatively, and entertainingly. He has now composed an inviting introduction to selected, but nonetheless quite numerous, chapters of active mathematical research, and his book undoubtedly opens doors to engrossing mathematical concepts that are not easily accessible to the lay reader. ... Dr Bell has definite opinions on a large number of debatable questions, and he does not hesitate to state them vigorously. ... He is not a uniformly reliable guide on more philosophical questions concerning the foundations of mathematics.

**13.2. Review by: T A A B.**

*The Mathematical Gazette* **37** (319) (1953), 71-72.

Of those who write about mathematics for the intelligent non-mathematician, few are as lucid and precise as E T Bell, and none more stimulating. The present volume revises and amalgamates two earlier books, now out of print, The queen of the sciences and The handmaiden of the sciences. It is not a history of the subject, though there is much about the history of mathematics; rather it attempts to make plain to the layman the spirit of modern mathematics, its roots in the past, its sudden turns and spurts on the urgings of a Newton or a Gauss, its remoteness from the prevalent crude materialism and its extraordinary knack of becoming vitally relevant to material concerns. ... The author is justifiably pleased that the two earlier books were read by many non-mathematicians-lawyers, doctors, engineers, business men, writers-anxious to find the spirit of mathematics, and he is even more pleased to have given numerous young readers a glimpse of what lies beyond the school curriculum. The new book should be equally successful; there is just as much need for its stimulating vigour.

**13.3. Review by: Donald H Potts.**

*The Mathematics Teacher* **44** (5) (1951), 352-353.

The avowed purpose of this book is two fold: to give those non-mathematicians who remember enough of their elementary mathematics a better understanding of what modern mathematics is all about, and, secondly, to enable young students to catch a glimpse of modern mathematics. Certainly there is nothing more disquieting to a mathematician than to find some person holding the opinion that a mathematician is a sort of super accountant who is a whizz at calculating. This book could do much to dispel such a misconception. It would be especially suitable for collateral reading in the last year of high school or the first years of college for all students who hope to have some understanding of what mathematics is about and how it serves science. ... This book belongs in the library of every teacher of mathematics and every serious student of science. To both it could not help but be a source of inspiration and enlightenment, for Dr. Bell has the gift of presenting the pro found concepts of the subject with a minimum of technical difficulty. It has been remarked that Dr Bell is "perhaps mathematics' greatest interpreter." There is no perhaps about it.

**13.4. Review by: Angus Armitage.**

*Bulletin of the British Society for the History of Science* **1** (8) (1952), 220.

Professor E T Bell of the California Institute of Technology is already known to students of the history of science as the author of two stimulating books, 'Men of Mathematics', and 'The Development of Mathematics'. He has now rewritten two of his less-known popular works, published before the late war, and has combined them into a single volume designed to furnish the amateur with a broad picture of modern developments in the technique and philosophy of mathematics. ... the most interesting feature of the book is the bold attempt which the author has made to introduce the non-specialist to the modern abstract algebras in which the operations as well as the quantities are generalized. ... Professor Bell's book will serve a useful purpose in introducing students to a realm peculiarly difficult of access and in opening their eyes to vital ideas which have not yet found a place in the curriculum. Not a few of its readers may be moved to seek for fuller information from more august sources.

**13.5. Review by: A J H M.**

*The Incorporated Statistician* **3** (3) (1952), 54.

This fascinating book, written by the Professor of Mathematics in the Californian Institute of Technology, can be recommended strongly to all students and teachers of mathematics who find it difficult to see the wood for the trees. The reviewer thinks wistfully of his own first year at the University, when such a book would have met a great need. It provides a bird's eye view of the subject, embracing most of the developments of the last hundred years such as groups, topology and Riemannian geometry. The reader must not, of course, expect to probe deeply into the intricacies of these subjects ... It contains many biographical details and is written in an easy style.

**13.6. Review by: Monica A Creasy.**

*Journal of the Royal Society of Arts* **102** (4929) (1954), 693-694.

This book has been written to try and convey to the layman something of the spirit of modern mathematics. It should be mentioned at the start that a fair amount of effort will have to be made to grasp the ideas the author is trying to put over, and some slight background of sixth-form mathematics, as well as enthusiasm, would be an advantage. The book is written in a lively style, and Professor Bell has probably succeeded as well as is possible in his attempt to write a popular mathematics. Some personal bias has inevitably entered into the book, and the reviewer's main criticism is the over-emphasis on algebra and geometry while there is hardly any mention of developments in function theory and analysis. ... One important criticism of this book is the failure to convey the fact that a large part of pure mathematics exists and moves forward independently of any applications it may have. One feels the author is constantly trying to 'justify' pure mathematics by giving examples of its uses ; in this sense 'the spirit of modern mathematics' fails to come across. However, the book is well worth reading, and the author's frequent historical references give a pleasing sense of the continuity of the development of the subject.

**13.7. Review by: Frederick G Graff.**

*Amer. Math. Monthly* **58** (7) (1951), 502-503.

This book is a thorough revision and an amplified integration of two popular volumes of mathematics, The Queen of the Sciences, 1931 and its sequel, The Handmaiden of the Sciences, 1937 by the author. Here is a fascinating account of selected topics from the huge accumulation available in the developments in pure and applied mathematics from the geometry of Euclid to the most recent findings in mathematical physics. ... Informally written and characterized by the author's provocative and stimulating style, mathematical ideas are presented and evaluated to suggest that mathematics is vigorously alive, is still growing, and is indispensable to an understanding of some sciences and technologies and for a deeper appreciation of the philosophy of science. This theme is the greatest virtue of the book.

**13.8. Review by: Edith Ruth Schneckenburger.**

*Mathematics Magazine* **26** (3) (1953), 171-172.

As in his previous books the author's informal style creates the impression that he is conversing with the reader. One appreciates the occasional touches of humour. This book can be read with understanding by the layman for whom it is intended. No doubt it can be read with greater appreciation by students of mathematics. It should be recommended especially to teachers of mathematics or physical science and to undergraduate majors who are planning to become teachers or research workers in these fields.

**14. Mathematics: Queen and Servant of Science (Reprint) (1987), by Eric Temple Bell.**

**14.1. Review by: Jeremy Gray.**

This book, first published in 1951, is now republished in an edition which will take it into its 40s. It covers a remarkable amount of ground: rings, lattices, matrices, groups, Galois theory, number theory, the discovery of Neptune, the calculus, Fourier theory, questions of the infinite in mathematics, mathematical truth, and logic all get chapters. This must have been refreshingly novel when the book first came out; even now, when the "new mathematics" is giving way to computing in schools, it is enjoyably up-to-date. Indeed, the school diet is made all the more palatable for being shown here to be a version of the real thing. It is hard to think of another book which covers the same material, very hard to think of one which sticks so resolutely to its task of telling you what the mathematics is and why it matters. Indeed, one is struck by how many of Bell's illustrations have become part of the general knowledge we all have about our subject. It may be that this is where many of us first learned those scraps of information that quickened our interest and which we like to relay to our students in our turn. It is still a good book to put in front of anyone a year or two away from leaving school who wants to know more about mathematics. Bell's knock-about style contributes to the fun, although one would not want to see too many imitators. Some criticism can be raised. There are some historical inaccuracies ...

**15. The Last Problem (Reprint or 1961 edition) (1990), by Eric Temple Bell.**

**15.1. Review by: Barney L Erickson.**

*The Mathematics Teacher*

**84**(6) (1991), 489.

This is an excellent book! Bell has done an outstanding job of putting a historical perspective behind Fermat's last problem. For anyone wanting to obtain a perspective of the personalities and cultures of the people who were instrumental in laying the groundwork for Fermat's work, this book is an absolute must. The book is excellent reading regardless of whether the reader is interested in Fermat. The research into the lives and cultures of people in mathematics is truly outstanding. Anyone reading this book can't help but have a better understanding of the problems confronting the early mathematicians; the book reads almost as a novel (I enjoyed it more than many novels I have read). ... This book is a must for anyone who has an interest in the history of mathematics. Bell sheds a lot of light on many phases of the historical development of mathematics, especially the number-theory aspect.

**15.2. Review by: Bruce Carl Berndt.**

*Mathematical Reviews*, MR1075993 **(92b:01005)**.

"The last problem" refers to Fermat's last theorem. Readers expecting a historical account of the problem and efforts to prove it in the 297 pages written by the author will be very disappointed; the author does very little more than just state the problem. However, Lehmer, in a four-page "aftermath", and Dudley, in four more pages of notes, provide solid mathematical history and information about this famous problem. Supposedly, this is a book on the history of mathematics and mathematicians connected to Fermat or the "last theorem"; the connections are often tenuous. ... The history and biographies are permeated by the author's opinions and judgments. Often the evidence for his uncharitable pronouncements is thin. Those who wish to learn history in an entertaining fashion will enjoy this rambling account.

JOC/EFR January 2015

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