1.1. Purpose of the Book: B L Van der Waerden.

The "abstract," "formal," or "axiomatic" direction, to which the fresh impetus in algebra is due, has led to a number f new formulations of ideas, insight into new interrelations, and far-reaching results, especially in group theory, field theory, valuation theory, ideal theory, and the theory of hypercomplex numbers. The principal objective of this book is to introduce the reader into this entire world of concepts. While, for this reason, general concepts and methods stand in the foreground, particular results which properly belong to classical algebra must also be given appropriate consideration within the framework of the modern development.

1.2. Review by: Saunders Mac Lane.
Notices Amer. Math. Soc. 44 (3) (1997), 321-322.

B L van der Waerden's early studies in the Netherlands of algebraic geometry led him to think about useful definitions of intersection multiplicity for curves and surfaces. He heard that Emmy Noether at the University of Göttingen used newer ideas about ideals to provide precise definitions of such multiplicities. So he went to Göttingen to listen to her inspired but sometimes confusing lectures on ideal theory and presently gave a very clear course of lectures on ideal theory. He then visited the University of Hamburg, where Professor Emil Artin (in 1921) was giving his impassionately insightful lectures on modern algebra. For a brief period there developed a plan that Artin and van der Waerden would collaborate to prepare an algebra text, but Artin did not get around to writing up his planned chapters. Then van der Waerden proceeded alone to write and publish (with Springer) his two-volume 1931 text Modern Algebra, with the caption "using lectures by E Artin and E Noether".

This beautiful and eloquent text served to transform the graduate teaching of algebra, not only in Germany, but elsewhere in Europe and the United States. It formulated clearly and succinctly the conceptual and structural insights which Noether had expressed so forcefully. This was combined with the elegance and understanding with which Artin had lectured. The first volume included his neat and clean presentation of the Galois theory, a presentation which rapidly replaced the earlier, often obscure treatments. The volume also covered formally real fields and valuation theory. The second volume covered ideal theory, algebraic integers, linear algebra, and representation theory. The whole was inspired by a facility for conceptual clarity and was written in simple, understandable German. Upon its publication it was soon clear that this was the way in which algebra must now be presented. Its simple but austere style set the pattern for mathematical texts in other subjects, from Banach spaces to topological group theory. When I first taught modern algebra as a beginning instructor at Harvard University in 1934, I of course used van der Waerden as my text.

1.3. Review by: Daniel Zelinsky.
Bull. Amer. Math. Soc. 57 (1951), 206.

The two volumes in English are a translation of the second German edition (1937) with the exception of some additions and revisions by the author in vol. 1 on the subjects of polynomials and valuations of algebraic number fields. The translation of vol. 2 is very readable. Unfortunately the translation of vol. 1 suffers from clumsiness of English style, some literal translations of German terms where standard English terms would be preferable or necessary, occasional inconsistencies, and some simple errors. The reader would be warned especially about chap. 10 which is full of too literal translations, particularly the erroneous use of "perfect" for "complete" - and this error is not made consistently. In any case, van der Waerden is now available to the student who reads only English, though he will miss some of the lucidity in vol. 1.

2.1. Review by: I Bernard Cohen.
The Scientific Monthly 80 (6) (1955), 377.

This well-written, beautifully printed, handsomely illustrated book will prove of great value to any one who is interested in the historical aspects of mathematics. The title should not mislead the reader into believing that Van der Waerden has included any material on "science," even "exact science" which might include astronomy and parts of physics such as optics. In this limited domain Van der Waerden has produced a work of exceptional merit, which is likely to remain the standard book in early mathematics for some time to come.

One of the most attractive features of this book is the wealth of factual material it contains. Major texts are quoted as well as analysed, so that the reader may follow the stages of interpretation with the control of the original material. This is especially important in the case of much of early mathematics because the texts themselves are often vague or fragmentary and can, therefore, easily give rise to conflicting interpretations. Another valuable feature is the care with which Van der Waerden always distinguishes between verified fact and opinion.
...

The author is a distinguished mathematician, especially well known for his work in modern algebra, who displays here and in his other recent historical writings a keen perceptive sensitivity for the development of mathematical thought. ... The wealth of information contained in this book, so admirably presented, should provide a corrective to many misinterpretations now current and should attract a host of readers who will share with the author the fascination of the growth of exact thought.

2.2. Review by: George Sarton.
Isis 46 (4) (1955), 368-369.

This book, first published in Dutch under the title Ontwakende Wetenschap, covers the history of mathematics from 3000 B.C. to the middle of the sixth century after Christ. A tremendous subject which could not possibly be covered in 300 pages, except if it were limited in one way or another. The author's method is to restrict his account very largely to the mathematical contents.
...

Van der Waerden's book is now undoubtedly the best guide for the study of the history of ancient mathematics. It is beautifully illustrated, but it is a pity that many of the illustrations are so-called portraits of ancient mathematicians. Each of those portraits is a lie, not a little lie but a very big one, for none of them can be authenticated.
...

The title Science Awakening puzzles me very much, for the book describes in detail a process covering thirty-six centuries and tells the whole evolution of Egyptian, Babylonian and Greek mathematics, three complete cycles involving awakening, growth, "golden age," senescence and death. Moreover, he speaks only of mathematics and its applications, not of any other branch of science, not even astronomy. We cannot have positive knowledge concerning the first awakening; we can only guess, but is it not plausible that science developed in the medical direction before the mathematical? Does the word "awakening" apply to an account of mathematics developed far enough "to satisfy the demands of stricter logic" and "brought to the state of perfection, beauty and exactness"? That is not "awakening" but full maturity, and to use the author's word "perfection." Why use a cryptic and misleading title instead of the true one "History of ancient mathematics"?

2.3. Review by: M H Stone.
Journal of Near Eastern Studies 15 (1) (1956), 56-58.

In the book, Professor van der Waerden is concerned not only with writing from an original point of view a clear, well-documented and quite complete account of what is known today about ancient mathematics, but also with the causes underlying the course of its development and with its significance for modern science. A good part of what he has to say upon these aspects of his subject is to be found in the preface and in the chapter on the decay of Greek mathematics. After all, however, both the beginning and the end of the development of Greek mathematics remains shrouded in mystery. We are unable to trace in a detailed way the transmission of the mathematical lore of Babylon to early Greece; we do not know how to explain the creative sparks which kindled and spread the glorious four hundred year flaring up of Greek mathematical imagination between 600 B.C. and 200 B.C.; we are at a loss to understand the inner reasons for the decline of Greek mathematics or its failure to inspire either the Romans or the Byzantines to fruitful mathematical activities. ... In summary, Professor van der Waerden has written a fascinating and stimulating book which can be recommended to anyone desirous of understanding the wonderful intellectual development it traces.

2.4. Review by: Alex C Aitken.
The English Historical Review 70 (276) (1955), 433-435.

There is a sub-title on the dust-cover, Egyptian, Babylonian and Greek Mathematics, which does not appear on the title-page proper. With due respect to the evident intention of winning a wider public by a more popular title, it may be said that the sub-title more accurately describes the contents.
...

Chapter headings give some idea of the topics: the Egyptians, number systems, Babylonian mathematics, Thales, Pythagoras, the school of Plato, the Alexandrian era, the decay of Greek mathematics. This is therefore a compendious history of mathematics, with some reference to mechanics and astronomy, from about 2000 B.C. to A.D. 500.

Popular as the appeal of the work is meant to be, a casual turning of the pages will show a good many geometrical diagrams and algebraic examples; so that the reader will have to be sufficiently versed in mathematics to know something at least of the theory of numbers, of the solution of quadratic, cubic and biquadratic equations, the formulae for the areas and volumes of pyramids, cones, spheres and the like.
...

On the whole, for anyone wishing to have a view, mostly in general outline but occasionally in close detail, of the progress of mathematics over two and a half millennia, this book will provide a mass of information. It is interestingly diversified by plates showing cuneiform tablets, Greek vases, statuary, architecture and so on, though it must be said that these illustrations often seem independent of the text. It is a book less to be read through than browsed at in portions. The author sets out as one of his main purposes that of sending the reader to the originals or to translations of the originals, or memoirs elucidating these. In this respect he is likely to be successful.

2.5. Review by: S H Gould.
Phoenix 9 (2) (1955), 89-90.

This excellent translation of Volume 7 in the Dutch series Historische Bibliotheek voor de Exacte Wetenschappen will be welcome to all students of Greek civilization. Its subtitle is Egyptian, Babylonian and Greek Mathematics, but the chief interest of the Babylonian and Egyptian parts lies in the light they throw on Greek mathematics. As the publishers state, the volume is meant not only for mathematicians but for all educated people, educated in the sense that they can understand school mathematics and are interested in it.
...

It is impossible to deal in a short review with so vast a subject, countless details of which are of great importance for all subsequent mathematics. Perhaps a reasonably just impression will be given if we say that the author, who is professor of mathematics in Zurich and possesses a world-famous expository skill, has presented in attractive fashion a surprisingly large amount of the information contained in Heath's standard work on Greek mathematics. Similarly, the chapters on Egyptian and Babylonian mathematics are based on Neugebauer's Vorgriechische Mathematik and The Exact Sciences in Antiquity, in which connection it would have been interesting to find a fuller treatment of the remarkable mathematical consequences, so interestingly treated by Neugebauer, of the fact that the Akkadian and Sumerian languages are not etymologically related to each other.

2.6. Review by: E L White.
The Quarterly Review of Biology 30 (4) (1955), 429.

The Professor of Mathematics of the University of Amsterdam has written for the educated layman of Holland an early history of mathematics, which has been translated by Professor Arnold Dresden of Swarthmore College. It covers the development of mathematics from the dawn of history to the beginning of the Middle Ages, with special emphasis on the contributions of Babylonian empiricism to Greek rationalisation. The title is misleading; there were scientific pursuits other than astronomy, geometry and algebra, etc., which were awakening in this period. In short, there is not even a nod in the direction of biology. The book is beautifully printed and illustrated, and charmingly written, enough to hold even a biologist fascinated.

2.7. Review by: T A A Broadbent.
The Mathematical Gazette 40 (331) (1956), 61-62.

The author has taken pains to make his account as simple and as up-to-date as possible. He has made much use of Neugebauer's epoch-making work on the Babylonians, and of other recent researches. But he is not a mere compiler, for he presents a number of new ideas, particularly concerning Thales and Theaetetus. He also has something new to say about the decline of Greek mathematics; but here he seems to desert his own principles in order to follow Zeuthen in tracing this decline to technical reasons, such as the inadequacy of geometric algebra. These are surely not inner causes but superficial symptoms: the real reason is to be found in the failure of the Greeks and the Romans to effect a real marriage of minds. The Hellene recoiled from the pushing, practical Latin, who in turn despised the avid intellectual inquisitiveness of the Hellene; the consequent divorce of theory from practice entailed the ultimate sterility of both.

While this volume does not replace Heath's Manual, it makes a most admirable supplement. The Noordhoff typography is well up to its own high standard, and there are 28 magnificently reproduced plates; these, as a rule, do not directly illustrate the text, a function performed by numerous line blocks, but they serve to remind us of the amazing versatility of the Greek genius, that "love of beauty without extravagance" to which Euclid and Archimedes contributed as much as Aeschylus and Pheidias.

2.8. Review by: Jean Itard.
Revue d'histoire des sciences et de leurs applications 10 (1) (1957), 94-95.

This work is a translation of a work in Dutch published in 1950 by the same publisher, under the title Ontwakende Wetenschap. Egyptische, Babylonische en Griekse Wiskunde, of which our readers will find an analysis by E J Dijksterhuis, in volume III, n° 3 of our Revue (July-September 1950, p. 285).

The author and the editors, on the advice of H G Beyen, professor of archaeology at the University of Groningen, have almost completely renewed the illustration of this new edition. MM O Becker, O Neugebauer and a few others have suggested different additions and corrections.

The work, written by a respected mathematician, will remain a classic for a long time.

Here, in a study that is too brief, I would not like to mark the points where I do not accept the author's interpretations. They are numerous and, on the whole, relate to the attributions which are made of the works of Euclid to such and such an author of the fourth century. Because there is nothing left for this unfortunate man and it is too easy to conclude then that he was a good teacher, but without great mathematical genius.

Wouldn't it be better to admit that apart from his work we know nothing about him, and that there is no solid criterion allowing him to deny, or rather to attribute to some other, books like the 5th, the 10th, or even the 13th?

I also do not accept the derogatory judgment made on Archytas. Why attribute book VIII to him rather than to Theaetetus, if this existed? And why, after having rightly pointed out the logical weaknesses of this book, to affirm that the seventh is as for him of an impeccable rigour? Yet propositions VII, 6 and VII, 8 have no logical basis, as a careful reading can show. And Archytas, even if he was the author of the eighth book without being the author of the seventh, is as rigorous and as good a logician as any other valuable mathematician of his century.

But let us leave these questions of appreciation, where the arguments for and against are always very weak.

The mathematical analyses are always excellent, that of the 10th book of the Elements in particular. The documentation on Egypt and Babylon is very rich and the work will render invaluable services to the readers who will not yield too easily to the undeniable brilliance of the author.

2.9. Review by: Inkeri Simola.
Nordisk Matematisk Tidskrift 2 (3) (1954), 173-175.

The above-mentioned work is intended for anyone who is interested in the history of mathematics and gives an account of the latest knowledge of the mathematics of ancient times. It is based on careful source research - the author has become familiar with the content of old papyrus scrolls and cuneiform scripts as well as with preserved writings by ancient scholars.

At the beginning of his work, the author clarifies in a brief overview the importance of mathematics to our entire cultural development and demonstrates why the history of science is also the history of mathematics. As is well known, mathematics as a science originated during the era of Greek culture, and on that basis the area covered by the book covers the time from the very first stages of culture to Greek mathematics. The book ends with an examination of the maturity of this mathematics. The author himself says about his work:

The main purpose of the book is to show how Thales and Pythagoras based their teachings on Babylonian mathematics, but nevertheless gave it a completely changed, typically Greek character, how mathematics through Pythagoras' school and elsewhere reached an ever-higher development and gradually began to fulfil the strictly exact logic requirements and how mathematics thanks to Plato's friends Theaetetus and Eudoxos reached the stage of perfection in Euclid's Elements.

The author begins the history of the development of mathematics with a presentation on the culture of the ancient Egyptians and thus introduces us to their counting technique. So we see the nearly 4000 years ago, the Rhind papyrus drew examples of the so-called "aha-count". Then follows the mathematics of the ancient Babylonian cuneiform scripts, and we can see what astonishingly the Babylonians' art of calculation has already spanned. So, for example, the application of Pythagoras' teaching was already known to the Babylonians.

2.10. Review by: Alfred McCormack.
Scientific American 193 (4) (1955), 120-122.

The history of mathematics is harder to follow than the histories of other sciences. The language of course presents difficulties, but there is also the more serious obstacle that the origin of early mathematical ideas is rarely self-evident. Ancient theories of physics, of natural history or astronomy or medicine are understandable; we see how such notions evolved and we can fit them into the development of thought. Not so with the Babylonian sexagesimal system, the Egyptian methods of computation, the Greek concepts of proportion, the Pythagorean approach to numbers.

This book by a noted modern student of algebra attempts to explain the beginnings of mathematical science, to trace the roots of number systems and the practical arts of computation, to describe the emergence of algebra and geometry and of rigorously logical methods of proof. It is unquestionably the best published survey of Egyptian, Babylonian and Greek mathematics. Quoting at length from cuneiform texts, van der Waerden presents a remarkable picture of Babylonian accomplishments. The Babylonians solved equations with one and two unknowns and problems involving quadratics and cubics; they knew the summations of arithmetical progressions, they were able to find Pythagorean numbers; they understood proportionalities arising from parallel lines and the famous right angle theorem ascribed to Pythagoras; they devised the formula for the area of a triangle and a trapezoid, for the volume of a prism, a cylinder, a frustum of a cone and a frustum of a pyramid with square bases. And this by no means exhausts the list of their mathematical triumphs.

In his account of Greek mathematics, occupying three quarters of the volume, the author seeks to establish the connections between the Mesopotamian advances and the work of Thales, Pythagoras and their successors. here he offers conjectures and interpretations based upon slender evidence, a practice which he himself condemns elsewhere in the book in tilting at the conclusions of other historians. Still, a very strong case is built for the conclusion that Greek mathematicians were immensely indebted to older cultures.

The book is sometimes well above the heads of the general audience to whom it is addressed, but in spite of this and a number of other shortcomings it will richly repay a reader willing to give it his attention.

2.11. Review by: Oskar Becker.
Gnomon 27 (1) (1955), 48-49.

It is really very gratifying that the excellent history of Egyptian, Babylonian and Greek mathematics, which the author originally wrote in Dutch, is now made available to a wide audience through a good English translation. Since the work was already extensively discussed in this journal in 1951, only the new additions to the translation should be noted here. ...

2.12. Review by: Vera Sanford.
The Mathematics Teacher 48 (8) (1955), 573-574.

Science Awakening is the history of Greek mathematics with its antecedents in the mathematics of Egypt and Babylonia. The material is presented against the background of the civilizations that produced it. The illustrations have been chosen with that end in view and each has an explanatory note. Chronological summaries are given with each of the major divisions of the work so that the developments in mathematics and astronomy are keyed to political history and to the history of civilization. Throughout the book there are descriptions of the social conditions that fostered or at times retarded the development of exact science. In the past thirty years, much has been added to our knowledge of the mathematics of Babylonia by the researches of Dr Otto Neugebauer and others. In the preface, Professor van der Waerden states that he has subjected the theories regarding the mathematics of the ancient world to careful scrutiny and has rejected conjectures for which he finds no positive basis. One of these is the supposition that the Egyptians used a 3-4-5 triangle in laying out right angles. He has shown great care in indicating statements that are his interpretations, his own hypotheses. There is certain to be disagreement with some of them, but each is worth consideration.
...

Science Awakening is a happy combination of a scholarly treatment of the subject and its humanistic background, excellent illustrations, a beautiful format, and most important of all, the translation into English made by the late Professor Arnold Dresden of Swarthmore.

2.13. Review by: Duane H D Roller.
Books Abroad 29 (4) (1955), 479.

The last few years have seen the opening of a new and fruitful area of investigation in the history of science, namely the study of the mathematics of early Babylonian and Egyptian cultures. Necessarily, the results of this research have appeared in technical mono- graphs and papers, and thus the information has not been easily available to the nonspecialist. Under these circumstances this excellent history of mathematics, covering the period through the Alexandrian era and integrating the results of modern research on Babylonian, Egyptian, and Greek mathematics, is most welcome. Translated from the author's Ontwakende Wetenschap by Arnold Dresden, it is beautifully printed and well indexed.

2.14. Review by: Albert F Lejeune.
L'Antiquité Classique 24 (1) (1955), 263-264.

"Science Awakening" is an inadequate title. It is in fact a history of pure mathematics in antiquity until Proclus. Not only is there no mention of observational or experimental sciences, but even the history of the various branches of applied mathematics is not treated as such. Astronomy and geography hardly appear here except in so far as they contributed to the progress of the geometry of the sphere and trigonometry. Acoustics, optics and mechanics are practically neglected. This equivocation dissipated, we must declare that this history of mathematics is remarkable in every way. One of the main merits of the author is to resolutely reject the convenient myth of the "Greek miracle". Ever more precise knowledge of Egyptian and especially Babylonian documents makes the close dependence of Greek on Eastern mathematics more and more evident. The author is fully aware of this research, in which he has taken an active part. He takes care not to fall into this other excess which consists in attributing to the pre-Hellenic sciences, and in particular to Egyptian science, results which are not attested anywhere, subject to the completely gratuitous assumption that they would have remained the secret of a caste. Like O Neugebauer in his Exact Sciences in Antiquity (1952), van der Waerden starts from the comparison of Eastern and Greek sources and endeavours, from their confrontation, to determine what the Greeks drew from the Babylonians and Egyptians.
...

The presentation of the material is impeccable. The topics, judiciously chosen, are embellished with a lucid and lively commentary. Remarkable both for its clear and direct exposure of methods and its particularly interesting conclusions with regard to pre-Euclidean mathematics, the work seems to us destined to become one of the classics of the future.

2.15. Review by: Michael Hoskin.
Blackfriars 38 (447) (1957), 276-277.

It is always exciting to uncover popular fallacies, and in few fields can they be as numerous as in the history of science. What other study can offer canards to rival the almost universal conviction that the medievals thought the earth was flat, or the repeated assertion even among scholars that Galileo was the first to challenge Aristotle's dictum that bodies of different weights fall with different speeds? Professor van der Waerden's exclamation, 'How many fairy tales circulate as "universally-known truths!",' shows that he finds a similar fascination in the study of what the book calls 'antique mathematics', and his consequent scepticism is one of the most refreshing features of this work which despite its title is devoted largely to the pure mathematics of the Egyptians, Babylonians and Greeks. The author is a distinguished geometer, and those who are interested in the elementary but often intricate mathematics of pre-Christian times have at last a comprehensive, scholarly and truly mathematical work written with a disarming friendliness of style.

Unfortunately, an attempt has been made, by the addition of sketchy background introductions to various chapters together with a number of admittedly beautiful illustrations and some talk of the importance of pure mathematics, to turn this important mathematical study into something like a history of culture; hence the misleading title, and the sudden changes of style which occur.

2.16. Review by: Ph S.
Hommes et mondes 104 (1955), 622.

Here is a history book that will fascinate all those interested in Mathematics, the importance of which we know in the ancient world and more particularly in Greek civilization. The great interest of this work is undoubtedly that it makes the reader follow the evolution of mathematical thought through ancient texts, from the Egyptians to the Babylonians and to the Greeks. One will find there the old demonstrations, such as they reached us, and certain explanations which facilitate comprehension. So these are precise facts and not vague statements, which show here the genius of a Thales or an Archimedes. On the other hand, this solid work has the merit of bringing into the history of Mathematics new facts brought to light by the recent discoveries of cuneiform texts; thus, we will see, clearly established, the preponderant role played by the Babylonian mathematicians (whose names are unknown to us) and how in their wake Thales and Pythagoras gave Mathematics a different and specifically Greek character. Clear presentation means it is easily assimilated with basic mathematical knowledge. The material presented is worthy of the elegance of the demonstrations.

2.17. Review (of 2nd edition) by: Carl B Boyer.
Science, New Series 134 (3494) (1961), 1975.

The book remains a model of historical-mindedness, a model in which opinions are independently arrived at from a critical reading of the best sources and he knows perfectly well to be the authorities. The author, for instance, makes use of Neugebauer's recent valuable research on the role of Bablonian methods in the development of Greek mathematics; but whereas Neugebauer concluded that "the traditional stories of discoveries made by Thales or Pythagoras must be discarded as totally unhistorical", Van der Waerden argues that this research "knocks out every reason for refusing Thales credit for the proofs and for the strictly logical structure which Eudemus evidently attributes to him." Here and there a reader may question a bold conjecture made by the author. Were the "inner causes" of the "decay of Greek mathematics" really the difficulty of geometric algebra and the difficulty of the written tradition? Not necessarily; but then, one of the virtues of this book is that Van der Waerden distinguishes clearly between the historical evidence and the thought-provoking conclusions that he has drawn therefrom.

Science Awakening is as attractively printed as it is accurately written, and only the title is infelicitous. The work is a clearly circumscribed and well- ordered history of pre-Hellenic and Greek mathematics; and while mathematics may be a handmaiden of the sciences, it is not itself - at least in the usual sense of the word - one of the sciences.

2.18. Review (of 2nd edition) by: Edward Rosen.
The Classical World 55 (6) (1962), 168-169.

This is the second edition in English of the best single-volume history of ancient mathematics. It was first printed in 1950 in Dutch. Four years later the same publisher issued an English translation containing modifications, which were incorporated also in the German version (1956) made by the author's wife. This present English edition has now offered him a further opportunity to improve his book.

His extraordinary commercial success in a field in which the first edition of a work, however meritorious, is usually also its last suggests that a third English edition may materialize some day. In that event the author may wish to consider the following [errors].
...

These minor blemishes and the distressing typographical errors should be eliminated from any future issue of van der Waerden's masterly survey of ancient mathematics.

3.1. Review by: Joseph Agassi.
Science, New Series 157 (3790) (1967), 794-795.

Here are two excellent books in one, and yet a frustration to the reviewer. As a source book, this is an excellent collection of papers, most of the previously untranslated. As a history, it presents a new and highly interesting view of the rise of quantum theory, documented with new evidence which the editor has obtained by querying the principals, and with a new selection of papers to be viewed as prominent. It is frustrating because the author-editor imposes on the reader an agenda that is as good as a verdict. For the theory presented, namely that in the transition from the old to the new quantum theory nothing really served as a guideline but Bohr's correspondence principle, cannot be examined without reference to other topics. But this volume is devoted strictly to one topic. Other topics - wave mechanics, Hilbert space, Zeeman effect, spin and statistics, quantum field theory - have been excluded on various editorial grounds or relegated to a projected second volume. All the struggle, then, is finished in the two page preface in which these omissions are noted. The 59-page introduction that follows it is a brilliant historical study which throws new light on the selections presented in the volume. It should, sooner or later, be critically examined by other historians.
...

Van der Waerden has rendered a valuable service by putting the correspondence principle back on the map. even though he appears thus far as an apologist of one party in the dispute - the majority - rather than as a dispassionate observer; however, he can still correct this impression. It would be easier for him to do so, I suggest, if he stressed more the role that problems play in the advancement of science.

3.2. Review by: C V R.
Current Science 39 (5) (1970), 118.

Although Max Planck's famous lecture of 1900 gave quantum theory its essential form, the theory as he stated it was just the beginning. In this book, Professor van der Waerden collected 17 early papers which developed quantum theory into the form in which we know it. These papers appeared from 1917 to 1926, and were written by many of the leading physicists of the early 20th century. The collection begins with Einstein's "On the Quantum Theory of Radiation,'' an illuminating derivation of Planck's Law. Other important early papers by Ehrenfest, Bohr, Born, Van Vleck, Kuhn, and others prepared the way for the "turning point" in quantum mechanics. This crucial step is expressed in Heisenberg's paper ''Quantum-Theoretical Re-Interpretation of Kinematic and Mechanical Relations.'' 11 of the 17 papers in the collection are reproduced unabridged.

3.3. Review by: Charles F Hockett.
Scientific American 217 (5) (1967), 150.

Professor van der Waerden ... reprints (all translated into English if needed) 15 key papers, from Einstein's in 1917 on induced and spontaneous photon emission to the papers of January 1926 in which Wolfgang Pauli and P A M Dirac independently carry out the new matrix programme of the Göttingen experts as applied to the hydrogen spectrum, even in external fields. (Pauli's paper convinced most physicists that Quantum Mechanics is correct.)

In a brief but original and knowing introduction, studded with letters from Werner Heisenberg written to Professor van der Waerden in reply to explicit questions, and with much other published and unpublished testimony, a first effort at a history is presented.

The more personal tales are not neglected. Pauli, as usual, spoke caustically to the experts at Göttingen, who wanted "to spoil Heisenberg's physical ideas by your futile mathematics." Heisenberg had his 1925 idea for the matrix mechanics while on vacation, spent on the grassless isle of Heligoland in order to escape the pollen of Göttingen in June. "It was very late at night. I calculated it out painfully," he writes, "and it agreed. Then I climbed onto a rock and saw the sunrise and I was happy."

4.1. Review by: Robert H Berk.
Journal of the American Statistical Association 65 (332) (1970), 1680-1682.

This book is intended to provide an introduction to mathematical statistics. Although it can profitably be read by someone who already has some basic knowledge of the subject, the reviewer hesitates to recommend it as a text for the uninitiated. The reason is mainly that the book draws no clear distinction between probability and statistics. The introduction of statistical notions is so casual in places as to be almost invisible. No formal definitions of tests, estimators or confidence intervals are given. Instead, they make their initial appearances as illustrations of probabilistic concepts.
...

To the book's credit, believable examples appear here and there. One tired of the overworked agricultural data might find some useful ideas here. The author thoughtfully provides footnotes for references. A collected list of references would have been additionally helpful. To recapitulate, one who has some mastery of the subject could profit by reading this book. The viewpoint is a bit different from most English language books and some of the material covered is unusual. For students, however, there are more careful and comprehensive English language presentations of mathematical statistics available.

4.2. Review by: Alan Stuart.
Economica, New Series 37 (148) (1970), 437.

This is a translation of a German introduction to mathematical statistics first published in 1956. Professor van der Waerden, who has made some notable contributions to the theory of distribution-free methods in statistics, has written a book with an individual flavour, in many ways more palatable than the machined taste of the majority of books at this level. Although it does not stray far outside the conventional topics, the mathematical language is more sophisticated than usual - a knowledge of Lebesgue integration is assumed and a 15-page chapter early on discusses briefly the necessary results on multiple integrals, Beta and Gamma functions, orthogonal transformations and quadratic forms. These tools make possible the careful, and occasionally rather long, proofs of the fundamental results. There is an early discussion of the empirical determination of distribution functions, and a very detailed theoretical account of chi-squared tests. Generating functions and cumulants are surprisingly absent and there is perhaps less emphasis on distribution-free methods than might have been expected from this author. The translation is excellent, and there are many detailed examples in the text, served by a most useful special index which classifies them according to their subject-matter (four relate to demography and econometrics). The absence of any exercises for the reader will matter less to the student than the price, which so far as I know sets a new record for an introductory book of this size. Even if they can't afford it, mathematically weaned undergraduates should persuade their libraries to buy it.

4.3. Review by: D V Lindley.
The Mathematical Gazette 55 (393) (1971), 355-356.

According to the dust-jacket "one might say that the aim of this book is to replace R A Fisher's Statistical Methods for Research Workers. Fisher developed many excellent methods ... but he was not always able to justify his results because the mathematical methods ... were not yet available. The present volume develops the most important methods with full mathematical justifications". This is a most laudable aim, and when the author is as distinguished a mathematician as van der Waerden the prospect is an exciting one.

To some extent one is not disappointed by the text. The style is excellent and the argument beautifully clear. The material assumes some mathematical maturity and would be suitable for British undergraduates in their final (or perhaps, second) year. Surprisingly enough there are many good illustrative examples which well illuminate the text in the best tradition of applied mathematics. The material could easily serve as a text for mathematicians to study statistics: it would not be so suitable for scientists, or the research workers to whom Fisher addressed himself.

However, there are some reservations. Let me illustrate by citing the example of the method of maximum likelihood. Fisher did not originate the method, but he did develop it enormously and derived heuristically important new properties of it. We now know that these results are not always correct and therefore it becomes necessary to describe exactly under what conditions they are true. This turns out to be surprisingly difficult (at least at the level of an undergraduate text) and there is a real need for a text that discusses maximum likelihood in a rigorous fashion. The present book does not do this: it evades the issue by referring the reader to Wald and Wolfowitz's seminal paper, now 20 years old. There are many gaps of this sort and one still awaits a text that is mathematically satisfying and does not duck the important issues.
...

The treatment is often original and refreshingly different from the usual run.

5.1. Review by: Victor E Thoren.
Isis 67 (3) (1976), 478-479.

Because this work is so different from existing discussions of ancient astronomy, it is hard to know where to begin in reviewing it. Those familiar with B L van der Waerden's first volume, entitled simply Science Awakening, will have some idea of what to expect, but even they are likely to be disconcerted by the extent of van der Waerden's departure from the orthodoxy of scholarship in ancient mathematical science. For what van der Waerden has done has been to attempt an actual history of the birth of astronomy - to take the bits of information gleaned over the past century and try to piece them together into a developmental picture of pre-classical astronomy.

Those who have wrestled with the existing expositions of early exact science will be aware of the episodic treatment accorded it under the prevailing ethic; they will probably also understand some of the rea- sons behind that ethic. Any doubts they might have had about the problems inhering in the construction of a synthetic account will surely be resolved by contemplating van der Waerden's struggle. Yet, it seems clear that the project was worthwhile, and even overdue. Historians have long subscribed to the synthesis as the highest form of history, and philosophers have now pretty well established the significance of its analogue, the theory, in scientific progress. Wherever the history of science falls in the spectrum of scholarly enterprise, it must surely be able to profit from "bold leaps of imagination" in the same way that other disciplines have, through the stimulation of progress at the research front or by the dissemination of specialised knowledge from the research front to the profession at large. In this latter respect, The Birth of Astronomy is likely to be considerably more successful than its predecessor, Science Awakening.

5.2. Review by: Bernard R Goldstein.
Journal of Near Eastern Studies 37 (3) (1978), 275-277.

Van der Waerden is one of the small group of scholars who have made fundamental contributions to our understanding of ancient astronomy, and he is also well known for his lively and engaging style in presenting mathematics. Indeed, Science Awakening I (Dutch edition 1950, English editions 1954, 1961) which concerned Egyptian, Babylonian, and Greek mathematics has remained a classic in the field. In addition to presenting some of the technical aspects of ancient science, the author defines a broader goal, for we read in the introduction: "We shall study Egyptian and Babylonian astronomy in connection with stellar religion and astronomy. In this way we shall avoid taking astronomy out of the historical and cultural context to which it belongs." The results, unfortunately, are very uneven, and the proposed synthesis is not persuasive.

5.3. Review by: Owen Gingerich.
Science, New Series 188 (4190) (1975), 842-843.

A sequel to his Science Awakening volume on the early history of mathematics (1961), The Birth of Astronomy describes Egyptian and Babylonian astronomy in substantial detail, thus furnishing the most readily accessible source for the mathematical methods of Mesopotamian astronomy.

The author's relaxed and personal style makes his book interesting to read, although English-speaking readers will be distracted by such names as "Platon," "Aristoteles," and "Apollon," an idiosyncrasy inconsistently practiced and not found in the first volume of the series. Although the typography is excellent, the 31 plates have generally been copied from halftones in the German edition rather than afresh from glossy photographs, the result being low-quality and inexcusably moiréed reproductions.

Van der Waerden's account provides a remarkable juxtaposition of the "hard" analysis of the exact sciences with "soft" speculation on the intertwining of astronomy, astrology, and astral religion. In this new version, the German original has been considerably rearranged to give greater prominence to his long chapter on cosmic religion. Briefly, he argues that the religious currents and ferment of the Middle East in the 1st millennium B.C. provided the cultural framework out of which astronomy arose. The astral aspects of the Mithraic cult and Greek Orphism furnished the background against which the zodiac became a significant astrological concept. Spurred by the collapse of the Assyrian empire in 612 B.C., these religions gained new adherents and influence; in van der Waerden's view, the fact that systematic planetary observations begin at about this same time is not accidental. In turn, the rapid development of computational methods made horoscopic astrology a possibility for the first time. Incorporated into Zoroastrianism, astrology provided a vehicle for the spread of Babylonian mathematical procedures first into Greece and later into India and Hellenistic Egypt.

5.4. Review by: J D North.
American Scientist 63 (5) (1975), 593-594.

In the first volume of Science Awakening, the second English edition of which appeared in 1961, van der Waerden dealt with Egyptian, Babylonian, and Greek mathematics. The second volume covers Egyptian and Babylonian astronomy. The English is a revised and somewhat simplified, though considerably extended, version of the German edition which first appeared in 1965. A number of colleagues have helped with the revision, and although some matters of unresolved dispute have inevitably been included, the finished work is a remarkable tour de force, taking the reader within sight of the front line of current historical re search while demanding at the outset nothing beyond an elementary knowledge of mathematics and astronomy.
...

the book has the vital character one expects from an author whose opinions are formed on the basis of familiarity with original sources. It is thoughtfully illustrated and respectably printed; but why is it not uniform in style with the 1961 volume?

6.1. Review by: Barry Simon.
American Scientist 63 (6) (1975), 697-698.

This is a translation with some modernization of van der Waerden's classic Die gruppentheoretische Methode in der Quantenmechanik, which together with the books of Weyl and Wigner had profound influence on the development of quantum mechanics. It is always nice to have a translation of an old friend, but I must say I am not happy with some of the modernization. The author seems to have some of the modern mathematics wrong (e.g. the second sentence on page 13), and he fails to come to grips with some of the modern issues concerning the mathematical problems of passing from a Lie algebra representation to a group representation. It is also rather strange to have a "modernized" discussion with no mention of SU(3). About the only readers I could recommend the book to are those interested in the history and those already familiar with the mathematical theory of group representations who wish an introduction to some of the applications to physics.

6.2. Review by: Alan Hopenwasser.
Bull. Amer. Math. Soc. 82 (3) (1976), 451-455.

It is probably futile for a mathematician to speculate on how much a physicist would benefit from a book; in this case it is also academic. Most physicists will undoubtedly continue to turn to Eugene Wigner's Group theory. Wigner's book, first published in German in 1931, and then in translation, revised and expanded, in 1959, is considerably longer than van der Waerden's book and much fuller in detail. Some mathematicians, on the other hand, might well prefer a book which explains the essentials of quantum mechanics and the applications of group theory without presenting a complete and detailed picture. Van der Waerden's book promises to meet this need; but, unfortunately, it fails.

The tenor of the whole book is set at the very outset: we are told that a pure state of a mechanical system is defined by a wave function which is a solution of Schrödinger's differential equation. We are not told what a pure state is physically; the statistical interpretation of the wave function is given only three lines; and no attempt whatsoever is made to connect Schrödinger's equation with physical reality. This spirit repeats again and again throughout the book; the physics is assumed, not explained. A mathematician who does not already know quantum mechanics will be unable to read this book without extensive supplementary readings. The reader of this book obtains no sense of the physical reality of the subject discussed. This is a bit ironic when one considers the heavy emphasis on spectroscopy in van der Waerden's book, for it was precisely spectroscopy which provided some of the best experimental evidence for the theory of quantum mechanics.
...

... the mathematician who wishes to learn quantum mechanics, and take advantage of his mathematical knowledge in doing so, must look elsewhere. The mathematician familiar with quantum mechanics and desirous of seeing how group theory can be applied to physics may benefit from and enjoy van der Waerden's book. However, the physics described by van der Waerden, atomic and molecular spectroscopy, can be done for the most part without group representations. It is in other subjects, such as elementary particles, where the dynamics is mostly unknown, and where, as a consequence, we must rely heavily on conservation laws and symmetries, that group theory has a truly vital role to play.

7.1. Review by: H S M Coxeter.
The College Mathematics Journal 16 (2) (1985), 169-170.

The chief aim of this fascinating book is to demonstrate that mathematical ideas were communicated between civilizations in Europe, Asia and Africa earlier than is commonly supposed. As preparation for this investigation, the author carefully describes radiocarbon dating, which was introduced in 1946 by W F Libby and corrected by later authors using the tree-ring method. Ancient skills can be observed in megalithic monuments built between 4800 and 2000 B.C. in Portugal, Spain, Malta, Brittany, Ireland, Scotland and England. The enormous stones in 'henges' such as Stonehenge and Woodhenge are seen to be located on circles, ellipses and other ovals, probably constructed by stretching a closed rope round one, two or three pegs. The required positions of such sets of three pegs indicate knowledge of Pythagorean triangles (3, 4, 5), (5, 12, 13) and (12, 35, 37) in the Neolithic Age. The author cites several reasons for this belief that the geometric and astronomical skill necessary for such architecture must have developed in just one centre and spread to both Egypt and western Europe.

Careful comparison of ancient texts seems to indicate a tradition of teaching mathematics of well-chosen sequences of problems and solutions, a tradition which seems to have somewhere in neolithic Europe and spread towards Greece, Babylon, India and China. fifteen examples of striking similarities between various ancient civilizations. In particular, shows that Babylonian and Chinese algebra must have had a common source.
...

This is a book which every teacher of mathematics should possess, not only for its revelation of unexpected historical connections but also for its wealth of worked examples. Surely it must be stimulating for students to see how these tricky problems were tackled thousands of years ago, before the invention of trigonometry.

7.2. Review by: W R Knorr.
The British Journal for the History of Science 18 (2) (1985), 197-212.

B L van der Waerden is a respected algebraist and historian of mathematics. His Moderne Algebra has been a mainstay in the teaching of abstract algebra for over five decades, while his Science Awakening (English editions 1954, 1963) continues to stimulate students of ancient mathematics. In recent years van der Waerden's research has moved in a very speculative direction, following leads opened up in a long series of articles by A. Seidenberg on the ritual origins of ancient mathematics and science. The book under review synthesizes and extends several of van der Waerden's own articles, in which he has argued a pre-Babylonian ancestor for all the ancient traditions of geometry and arithmetic. He now proposes that the primary tradition arose within the neolithic culture of Indo-European peoples who migrated into Central and Northern Europe in the 4th and 3rd millennia B.C.

The thesis, if true, would represent a finding of unparalleled significance for historians of early science. One is thus obliged to scrutinize its claims with particular care; for the eminence of its propounder will inevitably accord it widespread attention among both specialists and general readers of mathematical history.
...

... the strengths of van der Waerden's book - the perceptive mathematical analysis of results and techniques, the far-ranging coverage of fields and traditions, and the inclusion of materials (as on Hindu and Chinese) not yet easily accessible in standard discussions are overwhelmed by the weaknesses of his interpretive framework. These weaknesses stem from his failure to scrutinize the implications of his hypothesis of neolithic origins. All his evidence is compatible with the generally accepted and far more plausible view of independent Egyptian and Sumerian origins in the 3rd millennium B.C., followed by their development and transmission to the Hindu, Greek and Chinese, through an intricate pattern of cultural interactions extending over several thousand years.

8.1. Review by: David J Winter.
The American Mathematical Monthly 95 (8) (1988), 781-785.

In A History of Algebra, Bartel Leenert van der Waerden covers the period from Muhammad ben Musa al-Khwarizmi to Emmy Noether, that is, the period from about the beginning of the 9th century to the first third or so of the 20th century.

A History of Algebra is written by a mathematician for mathematicians. In it, van der Waerden emphasizes the development of the mathematics and the interrelationships among ideas coming from different mathematicians. He discusses what was or may have been the motivation for research in a particular direction, as well as other factors and influences. He shows what mathematics was discovered and what the work really was about by describing the mathematics and showing how it was done - sometimes giving the crucial ideas, sometimes sketching a proof or outlining a manuscript, sometimes including proofs themselves.

The style is at the same time informal and precise. Although substantial mathematical prerequisites constrain the potential readership, A History of Algebra should be of interest to a fairly broad spectrum of readers from the mathematical community.

For the working algebraist or the student of algebra, A History of Algebra should be particularly useful. Van der Waerden not only places very important parts of algebra in historical perspective, but also gives an excellent expository discussion of some of the mathematics itself. He covers many important topics at length and, throughout the book, gives sources for material covered elsewhere. Furthermore, he treats much material which algebra students normally do not see in their graduate training, material which is important in understanding how and why the mathematics was created but is not needed in the modern logical development of it. At the same time, such material can provide deep insight into the modern theory and thus both render the theory easier to understand and clarify its significance and potential future.

One notable asset of A History of Algebra is the fact that the subject matter is not at all confined to algebra. The author is concerned not only with the algebra, but also with the mathematicians who discovered it and with their other related work, with the surrounding circumstances and with the relations among their various discoveries. It was therefore practical and necessary to include a wide variety of interesting topics from geometry, topology, analysis, physics, astronomy and other subjects, which are well integrated into the discussion.

Van der Waerden's choice of material, and his balance between the emphasis on history and emphasis on the mathematics, is reasonable. At the same time, algebra is a field about which much more can be said than can be put in any one book. Accordingly, there is a need for other books covering other important topics and other aspects of its history.

8.2. Review by: J J Gray.
The British Journal for the History of Science 20 (1) (Jan., 1987), pp. 96-97

First the content of this necessarily selective book. In the first part van der Waerden con- siders Al-Khwarizmi, Tabit ben Qurra and Omar Khayyam; Algebra in Italy; the theory of equations from Viete to Abel; Gauss; Galois; and Jordan. In Part Two he looks at the origins of group theory and at the theory of Lie groups. In Part Three he looks at algebras: their discovery; their structure; group characters; and finally representations. So it is very much a mathematicians' book. It is written from their point of view, and with their interests exclusively in mind. You will only find out from it what various mathematicians of the past did, there is no rich sweep of contextualization. But that is not necessarily a bad thing, and this book has several virtues. It is almost unfailingly clear. The arguments presented are summarized with a deftness that isolates and illuminates the main points, and as a result they are frequently exciting. Since nearly 200 pages of it are given over to modern developments which are only now receiving the attention of historians, this book should earn itself a place as an invaluable guide. Its second virtue is the zeal with which the author has attended to the current literature. Almost every section gives readers an indication of where they can go for a further discussion. As a result, many pieces of information are here presented in book form that might otherwise have languished in the scholarly journals. Since one must be cynical of the mathematicians' awareness of those journals, the breadth and generosity of van der Waerden's scholarship will do everyone a favour.

This is not a book with a 'thesis'. It is not organized round a governing idea, and so it is impossible to summarize. And it is certainly not a book to devour at a sitting; rather, it is to be dipped into and savoured. But it does at times get close to dealing with a difficulty any historian will confront who wants to understand the development of modern mathematics; the surprising and deep interconnections within the subject. Here, for example, one gets an impression of just how deeply the theory of elliptic functions and the emerging theory of groups were intertwined. This enriches the discussion to be found in H Wussing's pioneering work The Genesis of the Abstract Group Concept, (1984). On the other hand, and it is much to be regretted, there is less of the personal reminiscence than one might have expected from one who was so intimately involved with the creation of modern algebra. But one notes from the publishing details that the author will be 83 this year. May we all find a lifetime's devotion to mathematics keeps us so intellectually fit.

8.3. Review by: Jean Dieudonné.
Revue d'histoire des sciences 40 (1) (1987), 141-143.

The famous author of Modern Algebra, one of the founders of modern "abstract" algebraic geometry, has always been keenly interested in the history of science. He has published numerous volumes and articles devoted to the history of mathematics and their applications in the various civilizations of the ancient world, where he has shown that his erudition is equal to his mathematical talent.

Shortly before the publication of this volume, Van der Waerden published another work devoted to geometry and algebra in ancient civilizations. The history of algebra described in this volume therefore begins with the Muslims of the ninth century AD and ends with Emmy Noether and her school around 1930. The author indicates that he did not seek to write the history of all parts of algebra during this period; we will indicate at the end of this report the most important parts which are not treated; perhaps the author reserves them for a later volume?
...

To complete a history of Algebra, one should include that of multilinear algebra (theory of invariants and tensors) and commutative algebra (theory of fields and commutative rings). Naturally, the author could not avoid speaking a little about the theory of commutative fields in the part concerning algebras; but it is regrettable that an overview of this theory does not appear in the book, where neither Steinitz nor the modern form of Galois theory is mentioned.

9.1. Review by: Des MacHale.
The Mathematical Gazette 78 (481) (1994), 83-84.

I remember, when I was a student, browsing in the library and coming across van der Waerden's Modern algebra in two volumes. Here was something different, something you could get your teeth into! I had been brought up on the beautiful but clinical Herstein's Topics in algebra and I had little knowledge of the classical origins of abstract algebra. Springer have now issued van der Waerden's book in two handsome volumes under the title Algebra.

I think I would find it difficult to recommend Algebra: volumes 1 and 2 as the principal textbook for an undergraduate course in abstract algebra because the style is perhaps too easy-going, but certainly any mathematical library worthy of the name should have van der Waerden on its shelves. In addition, these volumes should be in the possession of any algebraist in the same way that any music lover should own some Mozart.
...

In summary, van der Waerden's Algebra: volumes I and 2 are strongly recommended as virtually compulsory classical reading for anyone who wishes to come to grips with algebra.