1.1. Review by: S Verblunsky.
The Mathematical Gazette 20 (238) (1936), 159-160.

This book gives a good account of recent researches in orthogonal series, and may be recommended not only to specialists in that subject but also to those interested in trigonometric series and in linear operations. The book is interesting throughout. The authors have done much to simplify the proofs; the most striking characteristic of the book is the effective, indeed the triumphant, use of the theory of linear operations.

The first chapter contains some preliminary theorems (usually with proofs) on series, integrals and linear operations. A great part of this chapter might well have been placed half-way through the book. The second chapter explains the fundamental concepts of the subject. The attractive third chapter is devoted to orthogonal developments in $L^{2}$. There is an account of Schmidt's method of orthogonalisation; of the expression for the best approximation in $L^{2}$ by means of Gram's determinants; and of Müntz's generalisation of Weierstrass' approximation theorem.

In the fourth chapter, a number of special systems of orthogonal functions are considered. On the one hand, the systems of Haar, Rademacher and Walsh; on the other hand, the polynomials of Legendre, Chebyshev, Laguerre and Hermite. Concerning the systems of the second class, as only the most elementary properties are considered, I think that they might well have been omitted. The space thus saved could have been devoted to giving fuller proofs throughout the rest of the book.

The fifth chapter is concerned with convergence and summability almost everywhere. The next chapter contains an account of orthogonal developments in the spaces $L^{p} (p ≥ 1)$, and the spaces of bounded and of continuous functions. Particularly noteworthy is the treatment of "multipliers"... Proofs of the theorems of F Riesz and of Paley are given. Strangely enough the authors, who are usually fond of appealing to the properties of linear operations, do not appeal to the convexity theorem of M Riesz, which furnishes the simplest proof of the theorems in question. ...

The seventh chapter contains an account of lacunary series. The final chapter is devoted to bi-orthogonal systems, and to systems of relatively orthogonal polynomials.

The book contains many misprints. ...

Many passages in this book seem to cry aloud "Give us references, or give us proofs!" But they cry in vain. I cannot help suspecting that the peculiar system of reference which the authors use is responsible for this pathetic spectacle. ...

In taking leave of this book, let me thank the authors for a few weeks' interesting, if strenuous, reading matter. I look forward to seeing a second edition, with the misprints corrected, the obscurities elucidated and the language embellished.

1.2. Review by: Børge Jessen.
Mathematical Journal. B, thematic journal. B (1938), 68-69.

This book is devoted to the general theory of orthogonal series. ...
...
The production is everywhere very careful and elaborate, and some of the results are new. From a local patriotic point of view, one is happy about the space that Gram's beautiful investigations occupy in the book. On the other hand, the historical information with which the Preface begins seems very misleading; there seems to be no point of reference for which the authors want to see a line of development from Gram through Fredholm to Hilbert and Erhard Schmidt. Unfortunately, we probably have to content ourselves with the fact that Gram, with his recognition of the importance of the method of least squares for arbitrary orthogonal series and of orthogonalisation, was ahead of his time, and reconcile ourselves to the fact that it is not due to his work that the development with Hilbert's and Erhard Schmidt's investigations were successful on this basis.

1.3. Review by: J D Tamarkin.
Bulletin of the American Mathematical Society 44 (1.P1) (1939), 20-21.

The present volume of the excellent Polish Series is devoted to the theory of general orthogonal functions of a single real variable. Desiring not to increase the size of the volume without proportionally increasing its usefulness, the authors omitted almost completely the theory and applications of special orthogonal functions including that of orthogonal polynomials, and concentrated their attention on general orthogonal functions as a tool in pure mathematics. Even in this restricted field no claim is made for "encyclopaedic completeness." Despite these somewhat severe restrictions the authors succeeded in presenting a very interesting material widely scattered in the literature, including also some new contributions of their own.

The book consists of eight chapters followed by a bibliography containing 129 items. Chapter 1 gives a brief exposition of general notions of abstract spaces, and linear operations and functional which serve as a most important tool in the subsequent developments. Chapter 2 introduces the fundamental concepts of orthogonality, completeness, closure, and best approximation. Chapter 3 discusses general orthogonal series in $L^{2}$ including theorems of Müntz and of Riesz-Fischer, and Parseval's identity. Chapter 4 treats of various examples, with particular attention given to orthogonal systems of Haar and of Rademacher and to the "probabilistic" interpretation of the latter. Chapter 5 is devoted to the theory of convergence (almost everywhere, un-conditional, ...), divergence, and summability of orthogonal series. The climax of this chapter is reached in an elegant proof of the fundamental theorem of Rademacher-Menchoff. A systematic use of "Lebesgue's functions" associated with orthogonal expansions deserves a special mention. Chapter 6 deals with orthogonal expansions in various spaces $(L^{p}, C, M)$ different from $L^{2}$. Among various topics treated here we mention the relationships between the closure and completeness of an orthogonal system; the theorems of Young-Hausdorff and of Paley; the theory of "multipliers" transforming orthogonal expansions of functions of various classes into each other; and a discussion of various singularities which occur in orthogonal expansions. Chapter 7 reveals various remarkable properties of "lacunary" series. Chapter 8, the last chapter, is of somewhat mixed character, being devoted partly to biorthogonal expansions, and partly to polynomials orthogonal relative to a given weight-function.

The exposition, which is in general clear and concise, in some places shows a tendency to be either somewhat vague, or so condensed that it will be difficult to follow for a reader who is not well versed in the field. The number of misprints (in addition to those mentioned in a list of 16 Errata) and of slips of pen or thought is not entirely negligible. ...

In order to avoid footnotes the authors are using a new scheme of cross references, which, according to the reviewer's experience, does not represent an improvement over the customary system.

2.1. From the Publisher.

Designed to present small mathematical challenges in everyday life. Only the first edition, printed in Poland with English and Polish issues contains the loose objects for demonstrations the reader can do. There are also 180 text illustrations some designed to show three dimensional effects with the two-colour glasses.

2.2. To the Reader.

The present book is not meant to be scientific in the awe-inspiring sense of the word; its purpose is merely to draw your attention to some objects which may have escaped your special notice, even if you have seen them, and others which you have noticed without surmising that they had anything to do with mathematics.

You are to look first at the illustrations, and then read the text, which in most cases is very short. In many cases you will understand the illustration without any commentary; in others you will find a short explanation. If I ask you why? try to find out for yourself. There are many questions you can answer, even if you are a schoolboy, without any help, though I should not like to vouch for that in every case, for some of the illustrations lead to problems no grown-up person has so far succeeded in solving.

Please don't spoil the dodecahedron; it can be folded only by pressing the red face against the opposite green one.

The red-and-green spectacles enable you to look at the anaglyphs at a distance of 12 inches; don't touch the gelatine with your fingers. The crosses + indicate the points which are to face your eyes. Look first with the right eye only, then with the left, then with both closing them slightly.

The cards at the end are to be placed on a table to get the edges flush, and then laid on the palm of the right hand and put in motion with the thumb of the left. They give you four different moving pictures.

2.3. Review by: W D R.
The Mathematics Teacher 33 (1) (1940), 47.

This little book is equipped with red-and-green spectacles for looking at its anaglyphs, a dodecahedron, and a set of cards which handled skilfully give the operator four different motion pictures. It is a humorous, casual presentation of a number of mathematical facts, curiosities, and classical problems, with illustrations on every page and a light running explanation. It does not pretend to be a systematic text on any part of mathematics. Mathematical recreation is the main object of this fine book. It would be a good book to have in the mathematics classroom or for the use of a mathematics club.

Space does not permit us to give an elaborate description of the details, but one will find many interesting bits of material that will be of interest to pupils such as soap-bubbles, crystals, Möbius bands, polyhedra and the like.

2.4. Review by: Tomlinson Fort.
The American Mathematical Monthly 46 (6) (1939), 354.

Mathematical Snapshots is hardly to be classified under any of the usual categories. Indeed, the temptation is to make a new category - An Evening's Amusement - and to let this book be number one. The author uses pictures, figures, and models in addition to text to call attention to things of a mathematical or semi-mathematical character that are frequently overlooked or misunderstood. There is nothing like formal proof. The style is delightful, although there is an occasional lack of clarity, apparently introduced by translation. In fact, this book was more interesting to the reviewer than any book on so-called mathematical recreations that has ever come to his attention. Anyone, mathematician or what not, will enjoy turning its pages, looking at the pictures, and handling the models which are enclosed, in a cover pocket, also reading the text. It is recommended to all, especially to those who are fond of puzzles and have a flair for mathematical recreations. The person who has never seriously studied mathematics, but who did well in high school mathematics and has a sneaking notion that he would have done well in more advanced work, will be delighted. Angle trisectors, circle squarers, and all such will find in it profitable channels for their abilities. Introduce them to it.

2.5. Review by: E T Bell.
Science, New Series 89 (2307) (1939), 248-249.

The only way to review this beautifully made book is to describe its rich and extraordinarily varied content in some detail. It is mathematical recreations at a new level of simplicity, interest and unusualness, somewhat reminiscent of Lucas at his best, but less formal. Each page has one or more excellent illustrations, some in two colours, and a pair of coloured spectacles is provided for use with the anaglyphs. The pocket also contains a coloured self-folding dodecahedron and a set of cards for mathematical movies. Perhaps "visual mathematics" describes the general character of the recreations. Wisely, the author has refrained from attempting to teach anything, although anyone who can turn the pages without learning something must be singularly stupid. As in all good recreations, the concealed mathematics sometimes lies very deep. In this sense the book is scientific. But it can be enjoyed by anyone with a grammar-school education.

As the contents are so unusual, we give a partial summary of the topics touched so lightly and so effectively by the author (who, by the way, is a distinguished mathematician). We find: dissections of rectangles; noughts and crosses; the slide rule; chess problems, Euler's 36 officers, the 15 puzzle; musical scales; simple nomograms; the golden section, Fibonacci's sequence and phyllotaxis; tessellations; the triangle of forces; Peaucelier's linkage; anaglyphs; straight-edge constructions, roulettes, cams; Minkowski's lattice theorem; the limaçon, conics, the tractrix; space-filling curves; the regular solids, crystals, densest packing, soap bubbles; orthodromes and loxodromes; ruled surfaces; the resolution of cusps on skew curves; topological problems - unicursal patterns, the bridges of Königsberg, knots, Möbius' strip, existence of a bilateral surface with a knotted edge, the map problem for a torus; Pascal's triangle and the "board of fortune," the Gaussian distribution - amusingly illustrated by an experiment on digitalised frogs, which inspires the author to rechristen the normal curve "the frog-line"; the law of biologic growth, and finally, a sombre mortality graph for the U. S. in 1910. Scholarly historical and mathematical notes (184) conclude this most fascinating book. It should perform a genuine service by popularising mathematics.

2.6. Review by: Robert C Yates.
National Mathematics Magazine 13 (7) (1939), 351-352.

This is one of the most delightful books on mathematics to appear in many years. In my opinion it should be made available to all students of grade school, high school, and college, for pure enjoyment and for the sake of the stimulation it will undoubtedly afford. As the author says, "the book appeals to the scientist in the child and to the child in the scientist".

The underlying theme, and the only apparent one, is the presentation of theorems and bits of mathematical information, objectively and pictorially. In this the author and his small corps of assistants have indeed been successful. Although there is no system indicated in the selection of topics, quite frequently one discussion leads naturally to the next following.

Among the many features will be found chess problems, the mathematics of music, nomograms, the path of a billiard ball, continued fractions, tessellations, constructions of the compasses, perspectivity, roulettes, Minkowski's theorem, closed figures of constant breadth, conic sections, Sierpinski's curve that fills a square, regular polyhedra and space filling, soap bubbles, world maps, the conoids, topology, Möbius bands, the four-colour map problem, Pascal's number triangle, and the Gauss curve.

The ingenuity displayed in the drawings and photographs, which appear on every page, is remarkable. The variety of illustrations of the Gauss curve in nature, the composite photograph of the catenary and soap film, those of the Möbius bands, the knotted ropes of topology, the cusped shadow of a twisted wire, etc., all contribute to the unusual charm of the book.

The anaglyphs, to be studied with the red and green spectacles found pocketed in the back, are disappointing. They are quite indistinct and do not give the intended effect. In the same pocket will be found a set of cards which give animated pictures of the path of a projectile, the famous brachistochrone problem, the path of the earth about the sun, and the line motion generated by the internal rolling of two wheels. There is also a coloured dodecahedron which, believe-it-or-not, collapses neatly into the pocket. Another minor criticism should be directed to the midpoint construction of p. 38. Eight circles are used where only seven are necessary.

The publishers are to be highly commended on this excellent construction job. The book, printed in several colours, is evidently the result of great care and considerable hand work.

2.7. Review by: T A A Broadbent.
The Mathematical Gazette 23 (256) (1939), 422-423.

We must be grateful to Professor Steinhaus for deserting the abstract subtleties of real variable theory for a while to provide us with this delightful Mathematical Zoo. The metaphor is the author's own; in his preface he fears that his book may prove to be "too scientific for a child and too childish for a mathematician" but maintains that "mathematical objects are sometimes as peculiar as the most exotic beast or bird, and that the time spent in examining them may be well employed." I am quite sure that every teacher who reads this book will agree with the last sentence, at least as far as the mathematical objects which Professor Steinhaus exhibits are concerned.

It is somewhat difficult to describe briefly the contents of the volume. A very large number of illustrations with short commentaries is hardly an adequate indication of the wealth of material and beauty of treatment. A few samples may be mentioned: knots, a map on a tore needing seven colours, Möbius bands, loxodromes, soap-bubbles, polyhedra, crystals, the conic silhouettes, Crivelli's Annunciation and its perspective, tessellations, chessboards, noughts and crosses; these have been picked at random openings. But above all it is the beauty and charm of the illustrations, 180 in all, which make the book one to possess; black and white diagrams, coloured diagrams, anaglyphs, and superb reproductions of photographs. One item which should fascinate most children and most mathematicians is the model of a dodecahedron. The net is in two parts, joined by elastic; it lies flat in the cover-pocket but when taken out springs automatically into solid form; the faces are coloured to exhibit one of the two essentially different four-colour arrangements possible for the dodecahedron.

Teachers who have seen this book are enthusiastic about both its beauty and its stimulating effect. Author, printer, publisher, and the many helpers who worked at the production of the illustrations are to be most warmly congratulated on a splendid piece of work, for which ten and sixpence is an absurdly cheap price. School libraries should order at once.

3.1. From the Publisher.

Mathematical problems involving objects not ordinarily associated with mathematics are visualised and explained. There are problems having to do with chess moves, knots, maps, cutting a cake, and dividing an estate.

3.2. Review by: T A A Broadbent.
The Mathematical Gazette 35 (313) (1951), 210.

This new edition of the most charming of all the books of the "mathematics for everybody" type is very welcome. The possessors of the first edition must count themselves fortunate, however, since the new edition is slightly less attractive than the original. The layout is too Americanised for my taste, but this is a criticism of comparatively little importance. More serious regrets are that there is now no colour, the anaglyphs are replaced by photographs of models, the delightful self-unfolding dodecahedron and other gadgets are gone, and in some instances replacements have supplied poorer pictures, as, for example, the shadow-pictures of the conics. Austerity is no doubt largely to blame. Against this, the improvements are substantial: 295 diagrams instead of 180; an increase in the amount of comment; a more systematic and orderly arrangement.

The professional mathematician in Professor Steinhaus, expert in the austere abstractions of the theory of the real variable, must have been highly amused at the antics of his other self, revelling in models, pictures, maps, chessboards, shadows, puzzles and jokes. But if the "sandwich" theorem is a joke, it is serious mathematics as well: "it is always possible to cut a sandwich with a plane stroke so as to halve the bread, the butter and the ham." The young mathematician, attracted by the charm of the book, may well be stimulated by a theorem such as this to pursue his routine studies with an increased zest, and the volume ought therefore to be in every school library. But this ulterior motive, however important, need not be unduly stressed. The book is a magnificent store of visualised ideas, and for this reason alone it is one which mathematical shelves, library or private, should contain. Even at its present-day price, it is well worth the money.

3.3. Review by: Editors.
Mathematical Reviews MR0036005 (12,44e).

The first English edition was published in 1938 [Stechert, New York, N. Y.]. In this edition additions to the text have been made and the number of illustrations increased considerably. The anaglyphs of the first edition have been replaced by photographs.

3.4. Review by: Bryant Tuckerman.
The American Mathematical Monthly 58 (10) (1951), 708-709.

This book is a welcome addition to the literature of recreational and popularised mathematics. The author's purpose is "to visualise mathematics" by illustrated discussions of a miscellany of topics such as polyhedra, dissections and reticulations, various loci, geometric projections, and the topology of networks and surfaces; a few subjects are less pictorial. The number of illustrations - photographs and drawings - averages more than one per page.

In the items in problem form, the solution usually follows the problem without paragraphing. Readers may prefer to pause after the statement of the problem in order to work on the solution themselves. The discussion is at the level of the interested amateur, with more difficult proofs omitted, although such a question as the unanswered "how?" on p. 7 is more than an "exercise for the reader." References are given.

The first edition of this book was printed in Poland before the war. The present edition has a fifty percent increase in text and number of illustrations, and the binding is better. The anaglyphs (red -green three dimensional drawings) of the first edition have been replaced by more satisfactory photographs, and the separate gadgets have been omitted. The one real loss is that the self-erecting dodecahedron which popped out of the earlier hook is now only illustrated. Any gadgeteer will want to build one. Interesting new material in the new edition includes a counterfeit coin problem; the fair division of a cake or of an estate; curves and strategies of pursuit; the dissection of a square into unequal squares; and a dissected cube to be reassembled. This book should be pleasant reading for amateur and professional mathematicians, and an occasional source-hook for high-school and college mathematics clubs.

3.5. Review by: James R Newman.
Scientific American 183 (5) (1950), 56-58.

Mathematics presents certain difficulties because, as Sir Bernard Darwin has said about golf, you must learn it first and think about it afterwards. There are many of us who would like to understand the subject better, to get the hang of it beyond the mere ability to arrive at the same sum when adding a column from bottom up and top down. But the preparations required for this increase of knowledge often prove too arduous. One evinces good intentions by buying the latest popular book on mathematics and, if resolve holds, even makes an attempt to read it. The ensuing disillusionment is swift and drastic; it is soon discovered that the mystique of square roots and fractions is as disagreeably elusive as ever, that if these elementary matters are not mastered, further progress in the art is impossible, that the publisher's blurb to the effect that here at least is a way to glide painlessly to the summit is a monstrous deception. It is disconcerting to learn that a ready grasp of the intricacies of world affairs does not enable one to answer such questions as: If it takes three men seven hours to build a hen house and a half, how many hen houses can five men build in an hour and a quarter? And it is particularly humiliating to find that your 12-year-old son, who knows nothing about the designs of Mao Tse-tung or the Schuman Plan, regards the hen-house problem as tiresomely simple. There is of course no law of nature that a man's mind gets sharper as he grows older.

This book by Hugo Steinhaus offers consolation for these melancholy truths. In its seemingly haphazard way it affords, in exchange for a relatively small expenditure of reader effort, a remarkably spacious view of the subject. It is a book to stretch the imagination without unduly straining the mind; but this is not to say that if you enjoy grappling with difficult ideas you will be disappointed, for they are to be found here in abundance.

The first edition of Mathematical Snapshots was published in Poland in 1939 and has been out of print for some years. It was a delightful little volume. Besides its many handsome diagrams and striking photographs, it possessed a side pocket containing a collapsible, multicoloured dodecahedron (held together by rubber bands which made it self-erecting when removed from its hiding place), a set of motion-picture 'cards which when rapidly riffled displayed certain geometrical laws, a pair of red and green Cagliostro spectacles which conferred three dimensions on the book's several anaglyphs, and a few other equally ingenious gadgets. Dr Steinhaus' introduction was so modest and amiable as to disarm all criticism. "You are right," he said, "there is no system in this book; important things are omitted and trifles are emphasised. Many things do not deserve the name of mathematics, and the author himself does not seem to know what his aim really was in publishing his 'mathematical snapshots.' They are too scientific for a child and too childish for a mathematician." Still, this was excessively modest. For Steinhaus succeeded not only in serving up a repast of mathematical objects "as peculiar as the most exotic beast or bird," but his book, for all its grab-bag disorder and despite the fact that his morsels rarely more than tickled the appetite for the strange and wonderful, afforded an amazing display of the richness, the variety and especially the interrelatedness of mathematical thought. His snapshots had a dual role. They were often be beautiful and fascinating in themselves and from that standpoint it was unnecessary to ask what they meant. Yet they were also pictorial representations of purely abstract relations possessing universal validity. Thus they could illumine for the thoughtful reader something of the nature of intellectual process - how we are able to interpret the physical world and make coherent and useful systems describing its behaviour. The very mishmash quality of the book serves to carry out this purpose.

The new edition of Mathematical Snapshots, alas, omits the spectacles, the dodecahedron and the other props. Steinhaus' earlier apologia is replaced by a more formal and less informative preface. Moreover, the illustrations, though many new ones have been added, are somewhat smaller and therefore not as strikingly handsome as those in the volume printed by Ksiaznica-Atlas in Lwow. But the principal substance of the book has been preserved, and the text has been considerably expanded and improved.

Many of the old favourites among recreations are included. The secrets of the esoteric games of ticktacktoe and three-in-a-row are here exposed to vulgar view. The standard analysis is presented of the "Fifteen" or "Boss" Puzzle invented by Sam Lloyd, an oddment which for a number of years was the rage in France and Germany. (At one time it was found necessary to post a notice in the Reichstag forbidding the legislators to move the little squares in the Fifteen Puzzle while more serious matters were being considered.) Steinhaus says that the Fifteen Puzzle went out of fashion when in 1879 a mathematical explanation of it was published in the American Journal of Mathematics. Evidently he has not recently inspected the novelty counters of drugstores in American cities; it is my impression that more plastic Fifteen Puzzles than prescriptions are dispensed in some of these emporia. Anyone who is intrigued by chessboard puzzles (as distinguished from chess problems) and other recreations involving the so-called Graeco-Latin squares will find fresh, diverting material in Steinhaus. The great Euler, who was not merely mathematically omnipotent but omnivorous as well, is represented in these pages for his solution of the famous Seven Bridges Problem and the Problem of the Thirty-Six Officers. How is a delegation of six regiments, each of which sends a colonel, a lieutenant colonel, a major, a captain, a lieutenant and a "sublieutenant," to be placed so that neither in any row nor in any file will regiment or officer's rank be repeated? It is dispiriting to learn that this cannot be done, although it is possible to place 25 officers in the desired order. I do not mean to be unduly irreverent about the importance of this mathematical discovery: the solution undoubtedly represents some contribution to mathematical knowledge and, indeed, it turns out to have practical value in horticultural and genetic experiments. As Santayana wrote: "It is a pleasant surprise to [the pure mathematician] and an added problem if he finds that the arts can use his calculations, or that the senses can verify them, much as if a composer found that the sailors could heave better when singing his songs."

Without straining matters and with considerable imagination and skill Steinhaus shows the relations between rectangles, irrational numbers, falling dominoes and tunes; between tessellations, the drying mud of a river bed, the mixing of liquids, nomograms, slide rules, Lake Michigan, musical scales and the measuring of irregular areas and lengths; between soap bubbles, geodesy, the earth and moon, maps and dates; between squirrels, screws, candles, conic sections, acoustics, billiard tables, the nervous system, electrocardiograms and shadows; between the Platonic solids, crystals, bees' heads, detergents, billiard balls, a rhombic triacontahedron and th e method of proportional polling where three political parties are contesting an election; between polyhedra, the city of Königsberg, haystacks, knots, dovecotes, the Möbius band, doughnuts, the colouring of maps and the combing of hair; between Boyle's law, Pascal's triangle, frogs, college freshmen, bacteria, digitalis, the height of sunflowers and the total length of American railways. I cannot elucidate any further the links connecting the members of these assemblages except to assure that Steinhaus makes the most unlikely cousinships appear quite plausible.

For those who feel a book on popular science fails unless it makes the reader goggle, there is a fine sprinkling of goggle-making revelations. The highest known prime number, $2^{127} - 1$ = 170141183460469231731687303715884105727, can be got by putting two chessboards side by side, placing one grain (of something or other) on the first square, two on the second, four on the third - doubling through all 128 squares and then removing one grain from the last square of the second board. The number of grains remaining on that square will then be the large number written above. Although a curve is a one-dimensional figure and cannot fill space, it is nevertheless possible, as was shown by the Polish mathematician Sierpinski, to give the formula for constructing a space-filling curve that will fill a square. If you have 13 coins, one differing in weight from the others, three weighings will suffice to identify the counterfeit. (I should warn that this is not easy; in fact I remember the current rumour, when the puzzle was first talked about a few years ago, that Albert Einstein had been unable to solve it.) If three dogs are chasing one another, they are likely to be wasting their time unless they have heard of the circles of Apollonius or at least realise that the most efficient path of pursuit for all of them is along a logarithmic spiral. It takes a highly complicated machine to draw a straight line (no point in using a straightedge since you have no way of telling whether or not it is straight until you have drawn a straight line), but it is always possible to cut a sandwich with a plane stroke so as to halve the bread, the butter and the ham. (The latter problem has been most carefully considered in an article by A H Stone and J W Tukey on "generalised 'sandwich' theorems," in the Duke Mathematical Journal.)

This, I hope, gives the flavour of a book both entertaining and of intellectual merit. I hope it will have many readers. I can promise even those who regard all mathematics as a basilisk an agreeable surprise.

3.6. Review by: A Bakst.
The Mathematics Teacher 45 (2) (1952), 134.

This is a second edition of a book by the same title and the same author, published in 1939. It is a completely revised and enlarged edition.

This is a book containing a wide variety of collections of "curiosa" which would delight any mathematician. However, one must be warned that if he is a "mathematician" not of the "field officer" category (major or any other higher rank) he will not have an easy sailing. This is food for the generals.

On the other hand if one cares to spend a little time and effort in unravelling that which the author "takes for granted that the reader will understand," then he will be rewarded far above his humble expectations. This reviewer was particularly interested in ascertaining whether this book could be recommended to a classroom teacher as possible source material for enrichment purposes. He is delighted to report that the answer to this question is affirmative.

Unfortunately, there is not a table of contents. Still more unfortunate is the omission of an index. Thus, a reader will find some (unnecessary) difficulty when trying to locate some specific examples and illustrations which might be useful for some special classroom purposes. However, this should be construed as a minor criticism.

The topics treated in this book range from the simplest cases such as triangles, squares, games, rectangles, numbers, and tunes to solids (Platonic and Archimedean), geodesies, topological situations, and even the Jordan curve. This reviewer's opinion, and it is his only, differs from the opinion of the author that it is best to avoid (even the simplest) explanation. Nor does he believe that this book can be profitably enjoyed by anyone "with a knowledge of high-school algebra." For example, on page 27 he gives an expression for √2 in terms of a continuous fraction with a repeating denominator 2. How many high-school algebra courses treat the topic of continuous fractions?

This book is a masterpiece of printing, but, and this is not a reflection on the author whose "mother tongue" is not English, such is not the case editorially.

4.1. From the Foreword by Martin Gardiner.

"The sheep and the goat", said Abraham Lincoln, "are not agreed upon a definition of the word liberty". One suspects that the professional mathematician and the layman have a similar mis-meeting of minds over the word "elementary". Are the problems in this collection "elementary", as the book's title indicates? They are in the sense that the word is used in the section on "Elementary Problems" in The American Mathematical Monthly, that is, problems that do not call for a knowledge of calculus or higher levels of mathematics. But, if the average layman reading this book expects to solve most of its problems, he must know thoroughly his mathematics up to calculus. Above all, he must be able to think clearly and creatively.

Does this mean that readers of lesser skill and knowledge cannot read the book without enjoyment and profit? Far from it! Professor Steinhaus has given complete, detailed answers to every one of his one hundred problems. A layman who does nothing more than turn immediately to the solutions and then think his way through them will find himself learning painlessly, almost without realising it, an astonishing amount of significant mathematics. The best way to use this book, if you find most of the problems too difficult, is to take the difficult ones slowly, perhaps no more than one or two a day. If you make no headway with a problem, turn to the answer and read it carefully, several times if necessary, until you fully understand it. Do not skip the problem just because you encounter unfamiliar terms or procedures. Try to find out from other books or from friends what the terms and procedures are all about.

The literature of recreational mathematics is enormously repetitious. A great merit of Professor Steinhaus' collection is that the problems are not only top-drawer, but that most of them are relatively new, not to be found elsewhere. I know of no book of solved problems on this "elementary" level so refreshingly free of the old chestnuts that usually abound in collections of this type.

Another great merit of this book is that its problems often have elegant, sometimes totally unexpected, solutions. What a delight to discover, for instance, that Problem 97, which seems to deal only with relations between friends in a social set, actually involves the structure of the dodecahedron! And how beautifully clear certain confusing problems become after the author has drawn a relevant graph. I think particularly of the graph for that bewildering problem of the watch with two identical hands (Number 58), the diagram of the cyclist's minimum path for catching two pedestrians going in opposite directions (Number 86), and the startlingly simple graph that proves the constant length of a ribbon tied around a box in a certain way (Number 66). Many department stores, by the way, now fasten boxes with an endless loop of slightly elastic tape in just the manner given in this problem. Such a method is efficient for precisely the reason disclosed in the problem's answer.

The more skilful reader, if he enjoys puzzle-solving, will find that many problems in this book can be generalised in interesting ways. Consider Number 87, in which four dogs, starting at the comers of a square, chase one another simultaneously. What happens if the dogs start at the corners of an equilateral triangle? Of a pentagon? Can you find a general formula for any regular polygon? Similar questions arise in connection with Problem 73. After you have mastered the proof for the minimum network joining the four corners of a square, see if you can find the minimum network for the five corners of a pentagon. A very pretty problem in three-space is suggested here, and one I have not yet seen in print. What is the minimum network joining the eight corners of a cube?

Problem 33 provides a beautiful solution to a problem once proposed by Lewis Carroll. Warren Weaver, in his article on "The Mathematical Manuscripts of Lewis Carroll," Proceedings of the American Philosophical Society 98 (5) (1954), expresses it this way: "Can a billiard ball travel inside a cube in such a way that it touches all faces, continue forever on the same path, and all portions of the path be equal?" Steinhaus does not add the last proviso, but, if the ball's chair-like path is adjusted so that it strikes each side at a point of $\large\frac{1}{3}\normalsize$ distance from one edge and $\large\frac{1}{3}\normalsize$ from the other edge, the proviso is met. Each segment of the path has a length of $\large\frac{1}{√3}\normalsize$. This was worked out in 1960 by Roger Hayward, who illustrates Red Stong's "Amateur Scientist" department in Scientific American. (See Hayward's article on "The Bouncing Billiard Ball" in Recreational Mathematics Magazine, June 1962).

Several years ago, Professor Steinhaus proposed the similar problem of finding an equal-segment solution for a ball bouncing inside a regular tetrahedron. This also was solved by Hayward, early in 1963, and published in the "Mathematical Games" department of Scientific American in September, 1963. Are there such paths inside the other regular polyhedra?

Problem 34 (one of the few of the one hundred that is not new; Henry Dudeney gave it as Problem 146 in Modern Puzzles (1926]) suggests an interesting extension into four dimensions. This occurred to me one day when I stopped at a store window in New York to look at a colour reproduction of Salvador Dali's remarkable painting of the crucifixion, Corpus Hypercubus. (The original is owned by the Metropolitan Museum of Art, New York; reproductions are obtainable from the New York Graphic Society.) The figure of Christ hangs on a three-dimensional cross formed by eight cubes. Dali clearly had in mind an "unfolded" hypercube. The three-dimensional cross floats above a flat, two-dimensional checkerboard. Light from an infinite source casts a shadow of Christ on the unfolded hypercube, suggesting that the historical event is a projection of a transcendent event onto the space-time continuum of our world. Just as the cube can be unfolded in many ways to make a flat figure of six squares joined by their edges (a "hexomino"), so can a hypercube be unfolded in many ways to make a solid figure of eight cubes joined by faces. A cube has six square faces joined at twelve edges. Seven of these edges must be cut to unfold the squares. The hypercube has eight cubical hyperfaces joined at twenty-four faces. To "unfold" it into three-space, seventeen of these faces must be cut, leaving the cubes joined at seven faces. In how many ways can this be done? The problem should not be difficult for mathematicians who know their combinatorial way around in four-space.

Pictures of and additional details on some of Professor Steinhaus' problems can be found in the revised edition of his wonderful book, Mathematical Snapshots, (Oxford University Press, 1960). I can recall the delight, some twenty years ago, with which I examined the first edition of this book, printed in Poland in 1938. It was crammed with paper and cardboard gimmicks. Graphs drawn on transparent paper were tipped-in over pictures. An envelope at the back of the book contained such things as red and blue spectacles for three-dimensional viewing of certain illustrations; a cardboard model of a dodecahedron threaded with elastic to make it pop into solid shape; a cardboard torus coloured with seven colours, each bordering the other six; a packet of thirty-one cards that could be assembled in a certain order, then flipped at either end, on both sides, to make motion pictures of a bullet travelling a parabolic path, a planet moving in an ellipse, a circle rolling inside a larger circle to generate a hypocycloid, and a ball racing down the curve of quickest descent ahead of another ball on an inclined plane.

Hugo Dynoizy Steinhaus was born in 1887 at Jaslo, Poland, and trained in mathematics at Göttingen University, where he received his doctorate. At present he is professor emeritus at the University of Wroclaw (Breslau) and one of Poland's most distinguished mathematicians. He has published some 150 papers on pure and applied mathematics, edited mathematical journals, and received many mathematical awards. His interest in recreational mathematics is lifelong and unbounded. In the preface to the first edition of Mathematical Snapshots, he stated that the book's gimmicks and haphazard arrangements were designed to appeal "to the scientist in the child and the child in the scientist". "Perhaps", he concluded, "I have succeeded only in amusing myself".

The same spirit of play pervades this little book of problems. Dr Steinhaus can rest assured that Sto Zadan (as the book is called in Poland) will, like his previous book, both amuse and educate many thousands of kindred souls in the English-speaking world.

4.2. From the Preface.

This booklet is an answer to a challenge: a few years after the war the inadequacy of mathematical education in our high schools became evident to the staffs of universities and technological institutes. Some responsible people felt that a closer collaboration between mathematicians and school teachers could no longer be postponed. A few scientists were among those who did their best to stimulate interest in mathematics by means of elementary problems published in an educational journal. Here the reader will find one hundred elementary problems and their solutions. Some of them are familiar to students in high schools, but it was by no means my intention to provide the teacher with questions he could find in every textbook. I have tried rather to formulate problems suggested by geometry, often without classifying their mathematical background. As higher mathematics is not supposed to lie within the reach of the reader I was limited in my choice and this explains the small size of this collection. The solutions, however, are detailed enough to be understood by the teacher and by those of his pupils who are not afraid of thinking. Some of the solutions have been found by readers of the bimonthly journal referred to above - their names are printed on page 196 of the Polish edition. The last chapter has a few questions without corresponding solutions. There is an extremely important reason for such an omission for at least some of the thirteen items of Chapter VII: The author does not know the solutions; he hopes that his readers will try to solve some of them, thinking that their solutions are known, and that this mistaken view will enable them to succeed where the author has failed.

The "Hundred Problems" may help some freshmen discouraged by the difficulties of higher mathematics. Showing them elementary mathematics from another point of view, the "Hundred Problems" tries to bridge the apparent gap between "elementary" and "higher" mathematics. The book appeared first in Polish; I have been helped essentially by Dr S Paszkowski in the preparation of this first edition. The English version now presented to the reader is a revised edition. The translation into English has been accomplished by Mr Bharucha-Reid in collaboration with Miss R Czaplifska, Mrs J Smólska and Mr H Brown. The author is very much obliged to all the persons named above, especially to Mrs Smólska, who spared no effort to make this little book readable by the English-speaking public. There is also a Russian translation, of which 100000 copies have been printed.

4.3. Review by: Floyd D Strow.
The Mathematics Teacher 58 (3) (1965), 259.

The many who have read and enjoyed an earlier book by Dr Steinhaus, Mathematical Snapshots, will want to try their hand at solving this set of problems which ranges over a wide variety of topics. Included in the hundred problems are amusing conundrums in equations and inequalities, circles and ellipses, polyhedra and spheres, chess, sports, and homely affairs. Twisters with such intriguing titles as "The Wandering of a Fly," "How Old Is Mrs Z?" "Volleyball League," "The Tailor's Tape," "Student Debts," and "A Strange Social Set," are furnished - with solutions.

A bonus set of thirteen problems with such titles as "Triangle in a Triangle," "Division of a Circle," "Unlimited Chessboard," and "Radii in Space" is offered without solutions. The reason for no solutions - Dr Steinhaus does not have them and invites solutions from his readers. Do not start this book with the idea of working all the problems in one evening. There are many, many evenings of enjoyment for mathematics students of all tastes. The problems are, for the most part, the kind a student needs to think about for a time before he can come up with a solution.

4.4. Review by: H M Cundy.
The Mathematical Gazette 49 (367) (1965), 102.

Of the 176 pages in this book, 35 are taken up with the 100 problems, 6 with 13 more problems without solution (some of them unsolved), and the rest are occupied by the author's solutions and comments. This is therefore much more than yet another collection of problems: it contains a great deal of mathematics besides. The problems range from the very easy - no. 3 "Prove that $5^{5k+1} + 4^{5k+2} + 3^{5k}$ is divisible by 11 for every natural $k$" - to the very hard: e.g. no. 21 - "If a unit square is divided in any way into three parts, then there is always at least one pair of points $P, Q$ belonging to the same part with $PQ > 1.00778$." In this and other problems, a preliminary digestion of the author's Mathematical Snapshots will give some clue to his favourite trains of thought. Some of the problems are not too clearly stated; this may be the fault of the translation from the Polish, or the reader may be deliberately invited to clarify the conditions for himself. Geometrical and combinatorial types tend to predominate, and a number involve the classification of alternative cases. The techniques used are strictly elementary - no calculus is required - but a certain sophistication will be needed which will appeal more to the teacher than the sixth-former. The latter, however, would undoubtedly benefit from the occasional problem as an addition to his regular diet. The book should certainly be in the library, and would make a good prize, except that it is rather expensive for its bulk.

4.5. Review by: Editors.
Mathematical Reviews MR0157881 (28 #1110).

Translation from the Polish edition of 1958 with a foreword by Martin Gardner. The Polish edition has the review MR0090552 (19,828c) which is as follows. The problems are divided into 6 sections, with the headings: numbers, equalities and inequalities; points, polygons, circles and ellipses; space, polyhedra and spheres; problems practical and ... impractical; chess, networks, pursuit. Solutions are given in an instructive manner. There is an additional chapter on problems without solutions.

4.6. Review by: Frantisek Wolf.
Mathematics Magazine 40 (3) (1967), 156-157.

An elementary problem means here, a problem requiring in its solution only high school mathematics. Otherwise these problems require a creative mind and as such they demonstrate the beauty and the originality of mathematics. I wish that a mathematics teacher would forget sometimes the normal flow of day to day teaching and give his class a taste of the best in mathematics. It is not just learning a new recipe and then training its application by doing problems which are fundamentally alike. Here are problems which can serve for such demonstration. They do not fit any particular chapter. They have to be "carried" around for a few days before the right technique is finally found. If the teacher strings out the solution of such a problem for a week or two, devoting to it daily maybe 10-15 minutes, he can teach the student the drama of the solution of a nonroutine problem and the beauty of creative thinking. It creates motivation for many gifted children and calls to mathematics those precious few who have the intuition, the gift, and the passion for abstract exploration.

The problems are mostly surprisingly new and different from those found in the usual puzzle sections. Most of the problems have solutions given in the second part of the book which present a kaleidoscopic brilliant diversity. He who chooses to read the solutions instead of attempting to find them himself will get familiar with an astonishing variety of mathematical techniques. Sometimes the solution takes a totally unexpected turn. Reading that (a) everybody in the group was unacquainted with six others (b) everybody belonged to some mutually acquainted triplet (c) there was no group of 4 persons in which all members knew one another (d) there was no group of 4 persons in which nobody knew anybody (e) everybody belonged to a triplet not known to one another (f) everybody could find among the persons whom he did not know a person with whom he had no mutual acquaintance within the group. Who of the readers suspects that the author is describing the position of the faces of the regular dodecahedron?

There are problems on numbers which naturally extend the usual material taught in high school algebra on divisibility, irrationality, sequences, on polygons, circles and ellipses, on polyhedra and spheres.

This little book can be a true friend, if tucked away into your library and used at odd times as a special treat, to help you brush off the harassing routine of everyday from your shoulders.

4.7. Review by: J Bronowski.
Scientific American 210 (6) (1964), 135.

Problems in elementary mathematics are not necessarily elementary problems in mathematics. Steinhaus, known for his professional papers as well as for his entertaining popular book Mathematical Snapshots, has collected in this volume problems and puzzles of number theory, algebra, geometry, chess and other fields. Some are original, some comparatively new (at least in this country). All are ingenious, some are quite difficult, some are unsolved and perhaps unsolvable. One need not know higher mathematics to understand the statement of the problems, but even though fully worked out solutions are given, a considerable aptitude for mathematical thinking is often indispensable to following the game.

5.1. From the Publisher.

"What does a mathematician do?" someone once asked the author, and from that simple inquiry sprang this entertaining and informative volume. Designed to explain and demonstrate mathematical phenomena through the use of photographs and diagrams, Dr Steinhaus's thought-provoking exposition ranges from simple puzzles and games to more advanced problems in mathematics.

For this revised and enlarged edition, the author added material on such wide-ranging topics as the psychology of lottery players, the arrangement of chromosomes in a human cell, new and larger prime numbers, the fair division of a cake, how to find the shortest possible way to link a dozen locations by rail, and many other absorbing conundrums.

This appealing volume reflects the author's longstanding concern with demonstrating the practical and concrete applications of mathematics as well as its theoretical aspects. It not only clearly and convincingly answers the question asked of Dr Steinhaus but also offers readers a fascinating glimpse into the world of numbers and their uses.

5.2. Preface by Morris Kline.

This reprinting of the third, enlarged edition of Steinhaus's Mathematical Snapshots is more than welcome.

The book must be distinguished from numerous books on riddles, puzzles, and paradoxes. Such books may be amusing but in almost all cases the mathematical content is minor if not trivial. For example, many present false proofs and the reader is challenged to find the fallacies.

Professor Steinhaus is not concerned with such amusements. His snapshots deal with straightforward excerpts culled from various parts of elementary mathematics. The excerpts involve themes of sound mathematics which are not commonly found in texts or popular books. Many have application to real problems, and Steinhaus presents these applications. The great merit of his topics is that they are astonishing, intriguing, and delightful. The variety of themes is large. Included are unusual constructions, games which involve significant mathematics, clever reasoning about triangles, squares, polyhedra, and circles, and other very novel topics. All of these are independent so that one can concentrate on those that attract one most. All are interesting and even engrossing.

Professor Steinhaus explains the mathematics and his fine figures and excellent photographs are immensely helpful in understanding what he has presented. He does raise some questions the answers to which may be within the scope of most readers but the reader is warned that some answers have thus far eluded the efforts of the greatest mathematicians. Mathematical proof demands more than intuition, inference based on special cases, or visual evidence.

This book should be and can be read by laymen interested in the surprises and challenges basic mathematics has to offer. Professor Steinhaus is mathematically distinguished, and, as evidenced by the very fact that he has undertaken to present unusual, though elementary, features, is seriously concerned with the spread of mathematical knowledge. The careful reader will derive pleasure from the material and at the same time learn some sound mathematics, which is as relevant today as when the original Polish edition was published in 1938.

5.3. Review by: Sharon Whitton Ayers.
The Mathematics Teacher 76 (9) (1983), 698.

According to Professor Steinhaus, this book was originally conceived in response to a layman's question, "What does one do all day when one is a mathematician?" This reprinting of the third edition includes numerous snapshots and diagrams that prove immensely helpful in understanding many different topics from mathematics. Although some of the topics are from geometry, graph theory, probability, and topology, the explanations are for both the average citizen and the mathematician. The situations presented are unusual and intriguing; the applications with accompanying snapshots offer a note of relevance to mathematics in general. The examples given in this exciting book would enhance the lectures of high school mathematics teachers and college teachers of geometry and discrete mathematical structures.

5.4. Review by: I J Schoenberg.
The College Mathematics Journal 17 (2) (1986), 197-199.

I could not foresee in 1939, when I enjoyed the first edition of Steinhaus's Mathematical Snapshots, that 45 years later I would be writing a review of its third edition. The Oxford University Press should be congratulated for making this work again available; this is the more desirable because of the present neglect of geometry, which is usually exclusively analytic, and serves mainly as an introduction to calculus. But who learns about Dandelin's theorem on the plane sections of a cone of revolution? Exceptions are readers of the recent excellent Invitation to Geometry of Z A Melzak.

It was in 1939 and now again with the 3rd edition, that the Mathematical Snapshots is the only book in which nontrivial problems are described with a profusion of figures and photographs, while proofs are replaced by the occasional small word "Why?" The rich content defies description, and I will limit myself to mentioning only two subjects that I find particularly attractive.

A. The Euclidean Algorithm. The Greeks knew well that the Euclidean Algorithm may be used to find the largest common measure of two given segments. But it was from Steinhaus that I learned that this algorithm is nicely carried out by means of rectangles and squares. ...

B. The isoperimetric inequality ...

5.5. Review by: E A Maxwell.
The Mathematical Gazette 54 (388) (1970), 178-179.

Snapshots appeared in 1950, so this further edition is in itself an indication of merit. The preface reveals it as an attempt to answer the question, "What does one do all day when one is a mathematician?" The range is superb, and the illustrations really excellent. You should buy the book, if only for diagram 342:-

What you see on 342 is a white thread wound around 23 nails and showing a polygon of 23 sides with all diagonals. This is a proof that this polygon and all starred 23-gons form together a single closed curve. The winding took 45 minutes. The same trick would be impossible for a 24-gon and would take much more time for a 25-gon. (Why?) Nails excepted, the thread passes through no point more than twice.

6.1. From the Foreword.

This selection of works by Hugo Steinhaus contains 84 papers and articles from the total number of his 255 publications covering the years 1908-1980. Items included here, besides the presentation of the most important scientific achievements of the eminent mathematician, were chosen also to represent the possibly complete picture of his work and thoughts. Together with mathematical papers and the papers on application, some of his memoirs, polemics and programmatic talks were selected.

The chronological order of the papers in the main part of the volume will facilitate the Reader in following the development of scientific ideas, as well as exhibiting the variety of interests of the author. Excluded from the chronological order were the miscellaneous articles gathered at the end of the volume.

To preserve the authenticity of thought and style no amendments were introduced with the single exception of clearing obvious misprints. Papers originally published in Polish were translated into English by Marcin Kuczma and Piotr Paszkiewicz.

Selected works are preceded by an article on the life and work of Hugo Steinhaus, originally published in 1971 by Edward Marczewski, and by the bibliography of scientific publications by Hugo Steinhaus.

Hugo Steinhaus (1887-1972) was a Polish mathematician and educator. He helped establish the Lwow School of Mathematics; he is also credited with "discovering" mathematician Stefan Banach, with whom he gave a notable contribution to functional analysis through the Banach-Steinhaus theorem. Author of around 170 scientific articles and books, Steinhaus has left its legacy and contribution on many branches of mathematics, such as geometry, mathematical logic, and trigonometry. Notably he is regarded as one of the early founders of the game theory and the probability theory.

6.2. Review by: F Smithies.
Mathematical Reviews MR0819233 (87g:01065).

This volume contains a selection of 84 papers by the distinguished Polish mathematician Hugo Steinhaus (1887-1972); it was prepared by an editorial committee headed by Kazimierz Urbanik. The papers that originally appeared in Polish have been translated into English and thus made more generally available. The book is prefaced by an article on Steinhaus's life and work, by Edward Marczewski, and by a full bibliography of Steinhaus's publications, containing 255 items. The papers selected are on a wide variety of subjects, including trigonometric and orthogonal series, set theory, measure theory, functional analysis, probability theory, the theory of games, and topological methods in geometry; some of Steinhaus' popular articles are included, and some of his papers on applications of mathematics in biology, medicine and other fields; there are also obituary articles on Zygmunt Janiszewski (1888-1920), Leon Lichtenstein (1878-1933) and Stefan Banach (1892-1945).

7.1. From the Foreword by Kazimierz Dziewanowski (translated into English).

You hold in your hands a record of the memories of Hugo Steinhaus, eminent mathematician, a founder of the Polish School of Mathematics, first-rate lecturer and writer, and one of the most formidable minds I have encountered. His steadfast gaze, wry sense of humour (winning him enemies as well as admirers), and penetrating critical and sceptical take on the world and the people in it, combined in an impression of brilliance when I, for the first time, conversed with him. I know that many others, including some of the most eminent of our day, also experienced a feeling of bedazzlement in his presence.

In my first conversation with Professor Steinhaus, he attempted to explain to me, someone who never went beyond high school mathematics, what that discipline is and what his own contribution to it was. He told me then - I took notes for later perusal - the following, more or less. It is often thought that mathematics is the science of numbers; this is in fact what Courant and Robbins claim in their celebrated book What is Mathematics?. However, this is not correct: higher mathematics does indeed include the study of number relations but a welter of nonnumerical concepts besides. The essence of mathematics is the deepest abstraction, the purest logical thought, with the mind's activity mediated by pen and paper. And there is no resorting to the senses of hearing, sight, or touch beyond this in the exercise of pure ratiocination.

Moreover, of any given piece of mathematics it can never be assumed that it will turn out to be "useful". Yet many mathematical discoveries have turned out to have amazingly effective applications - indeed, the modern world would be nothing like what it is without mathematics. For instance, there would be no rockets flying to other planets, no applications of atomic energy, no steel bridges, no Bureaux of Statistics, international communications, number-based games, radio, radar, precision bombardment, public opinion surveys, or regulation of processes of production. However, despite all this, mathematics is not at its heart an applied science: whole branches of mathematics continue to develop without there being any thought given to their applicability, or the likelihood of applications. Consider, for instance, "primes", the whole numbers not factorable as products of two smaller whole numbers. It has long been known that there are infinitely many such numbers,
***
However, Hugo Steinhaus's recollections are to be read not so much in order to learn any mathematics - although one can glean from them interesting facts about what mathematicians have achieved. The main reasons for reading them are as follows: First, he led an interesting life, active and varied - although this is not to say that it was an easy one since the epithet "interesting" as used of life in our part of the world has often enough been a euphemism for experiences one would not wish on anyone. Second, his great sense of humour allows him to describe his experiences in unexpected ways. Third, his vast acquaintance - people fascinated him - included many interesting, important, and highly idiosyncratic individuals. And fourth and last, he always said what he thought, even though this sometimes brought trouble on him. Since he had no definite intention of publishing his writings, it follows that he was even franker in them. This truth-telling in response to difficult questions, this reluctance to smooth edges, not shrinking from assertions that may hurt some and induce in others uneasy feelings of moral discomfiture: this is perhaps the main virtue of these notes.

The following were the chief character traits of the author of these notes: a sharp mind, a robust sense of humour, a goodly portion of shrewdness, and unusual acuteness of vision. For him, there was no spouting of slogans, popular myths, or propaganda, or resorting to comfortable beliefs. He frequently expressed himself bluntly, even violently, on many of the questions of his time - for instance, questions concerning interwar politics as it related to education (even though he, as a former Polish Legionnaire, might have been a beneficiary of them), general political problems, totalitarianism in its Hitlerian and communist manifestations, and issues of anti-Semitism and Polish-Jewish relations. He said many things people did not like back then, and things they don't like today.

I believe that especially today, when our reality is so different from that of Steinhaus's time, it is well worthwhile to acquaint oneself with his spirit of contrariness and his sense of paradox, since these are ways of thinking that are today even more useful than in past times.
***
Steinhaus believed deeply in the potential for greatness and even perfection of the well-trained human mind. He often referred to the so-called "Ulam Principle" (named for the famous Polish mathematician Stanisław Ulam, who settled in the USA) according to which "the mathematician will do it better", meaning that if two people are given a task to carry out with which neither of them is familiar, and one of them is a mathematician, then that one will do it better. For Steinhaus, this principle extended to practically every area of life and especially to those associated with questions related to economics.

A particular oft-reiterated claim of his was that people who make decisions pertaining to large facets of public life - politics, the economy, etc. - should understand, in order to avoid mistakes and resultant damage, that there are things they don't understand but which others do. But of course such understanding is difficult to attain and remains rare.

Hugo Steinhaus represented what was best in that splendid flowering of the Polish intelligentsia of the first half of the twentieth century, without which our nation could never have survived to emerge reborn. This constituted a great impetus for good, triumphing over tanks, guns, and the secret police combined - a truly Polish strike force.

7.2. Review by: Marta Petrusewicz.
The Polish Review 40 (2) (1995), 230-234.

Hugo Steinhaus' Wspomnienia i zapiski is exceptional for a number of reasons: the length of the period covered, the intertwined modes of writing, the wisdom of the author, and the beauty of the prose. The time frame is eighty years, four-fifths of a century, from 1887 to 1968. The book is a memoir, an autobiography and a journal; the autobiographer doubles as a writer of memoirs, and from time to time may become a "diarist"; intimate journal may intrude occasionally upon the text. It is, like Chateaubriand's memoirs, simultaneously a chronicle of experience and a judgment of it.

The author's experience of life and history - in Austro-Hungarian (Polish) Galicia, Germany, Poland, and the United States - includes two world wars, two occupations, a revolution, a rebirth of a state, genocide, and communist rule. His is the testimony of a wise and trustworthy witness, sober and perspicacious intellectual and, as such, an invaluable source for historians. At the same time, Recollections is a biography, written by himself, of a great mathematician, a long-standing member of the cosmopolitan community of reason. Along with Stefan Banach and Stanislaw Ulam, Steinhaus was the founder of the Lwów mathematical school and later of the Polish one. He was a professor at the universities of Lwów, and Wroclaw, and visiting professor at many foreign ones.

As one scholar put it, had Steinhaus written only scientific papers, his world reputation would not have been any smaller. But Steinhaus did more than write of mathematics. He was a great populariser. His Kalejdoskop matematyczny [Mathematical Snapshots], ran through dozens of editions in numerous languages. His famed Fraszki [Aphorisms] were for years the delight of larger audiences. His Slownik racjonalny [Rational Dictionary] is a credo of a rationalist written by a man of letters who delighted in words and believed in them. A great scientist and a great humanist, Steinhaus in many ways reminds one of Albert Hirschman.

Recollections is composed in various interwoven modes. The memoirs written from a retrospective time and stance, are interspersed with what the author calls the "snapshots of memory'' or flashback recollections. A third form is that of diary or journal, written to the moment; rare in the first part of the book, it grows to become dominant in the last part. This creates an impression of a crescendo, powerfully climaxing in the absence of an ending.

It is clear that Recollections was written for publication, though, in communist-ruled Poland, Steinhaus could not know when or where. It was he who prepared it for the press. The last entries, short and hurried, transmit a sense of urgency. They were made in March 1968 when violent anti-semitic campaigns were unleashed by the government authorities, Prague Spring was coming to an end and clouds were gathering over Czechoslovakia. The last entry is from July 1968, one month before the invasion of Czechoslovakia. After that, only a note: "A few days ago I turned over to Ossolineum [the publishing house] the memoirs for the period 1887-1919, for publication ..."

Ossolineum did not publish the book - in 1968 it could never have made it through the censor - however, the Ossolinski Library preserved the manuscript. When Znak [The Sign] was allowed to bring it out years later, its author was already dead (Steinhaus died in 1972). The present edition by Aneks [Annex] is careful, well annotated, and includes a good index. One only regrets that the editors did not postscript it with a brief account of the manuscript's fata.

The first part of the Recollections is a kind of a portrait, with added elements of time and motion, of a world that is no more. Steinhaus grew up in the milieu of the progressive Jewish intelligentsia. His father was a merchant, his uncle Ignacy a politician. A relative, Marceli Frydman, was Director of the Vienna Opera. These Galician Jews had a strong sense of citizenship, of belonging and of responsibility: they bore arms and served in the civil guards. That Steinhaus joined Pilsudski's Legions was not an exception. Though he and his friends greatly admired the army, particularly the uhlans in their blue jackets and red pants, they were not militaristic "tough Jews." Hugo's maternal grandfather was a well known pacifist, and Steinhaus often had harsh words for the military (Hitler's is "the stupidest army in the world under the command of the worst criminals").

The town in which Steinhaus grew up was Jasło, on the eastern border of the Austro-Hungarian monarchy. How far it was from the stereotype of a Jewish shtetl! Turn-of-the-century Jasło had electric power plants and a railroad, petroleum extraction and a refinery, and a large and lively population of Jewish and non-Jewish workers and farmer/workers. Domestic industry was developing in Galicia. Its urban "pulse-beat" was quick.

The world of Jasło was full of variety, diversity and mobility. A lovely scene: little Hugo is sitting on a window sill waiting for something extraordinary to happen. And the extraordinary happens all the time: a peasant wedding passes, a Gypsy leads his bear on a chain, Christmas carol singers, Jews in their Purim costumes, and occasionally "phenomena of a supreme kind," for instance a travelling circus. Once, during imperial manoeuvres, thieves pulled down a peasant girl's skirt, exposing her bare buttocks to the highest authorities of the monarchy just a hundred feet from His Majesty. But "high culture" ran in Jasło as well. A music-loving Mr Teodor, for instance, arranged for a Beethoven concerto to be performed, full orchestra and all. Hugo studied at a public gymnasium, whose director was Rusyn and history was taught by a veteran of the 1863 Insurrection. He read avidly, from Polish modernists to the Scandinavian vanguard, all - and it was not a little - that made its way to Jasło. In 1906 Steinhaus encountered the writings of Karl Kraus - Viennese critic, writer and moralist - and, as he recalls, "found his reading for the next fifty years." He was attracted by both Kraus's moral standing and intellectual clarity, and his mastery of the language; Kraus's German, liberated from nineteenth century affectations, became "slim, muscular, light and powerful." Steinhaus says that he learned Polish from Karl Kraus ... . German, vice versa, he learned from reading Tom Sawyer in German translation!

Steinhaus studied mathematics in the cosmopolitan atmosphere of the University of Tübingen, and joined the faculty of the Lwów University. He recalls the brilliant milieu of Lwów mathematicians, gathered in the Scottish Cafe, jotting down problems and solutions in the so-called "Scottish Book." Some solutions would be rewarded just with a beer, others with a full dinner. (Stanislaw Ulam carried this custom with him to the University of California.)

What is the portrait of Hugo Steinhaus painted by himself? In a word, it is that of a rationalist and a sundial-maker. A man of great restraint, his memoir is often private but rarely intimate. His marriage (I remember admiring the beauty of seventy-year-old Mrs Steinhaus), the birth of their daughter Lidka, the latter's engagement to young Jan Kott, the intimate facts of life are barely sketched. How to reveal without exhibiting your intimate self is a problem that has taxed autobiographers since St. Augustine discovered that the self is a hard ground to plough.

Hugo Steinhaus is discrete but not cold and removed. He had a superb sense of humour, was interested in people and their vicissitudes, and had many friends. He traveled and he loved mountains. When Steinhaus "discovered" the Tatra Mountains in 1905, he delighted in his encounters with Hungarian gentlemen and Jews from Budapest, more Magyar than the Magyars themselves. This love for the mountains remained his for life. I accompanied Steinhaus on some day-long excursions, he seventy-five, I fourteen. A man of great sobriety, he was able to see clearly even in the midst of tragedy. When, in 1939-1940, his world went to pieces, he was not blinded by pity or nostalgia to this world's defects.

Fascinated with applicability of knowledge, throughout his life Steinhaus attempted to apply his work to economics and technology, to medical instruments and price policies, following what he called Ulam's rule: "a mathematician will do it better!" He was curious. One of the lasting motifs of Recollections is the building of an "introvisor," an instrument to look inside various things, the human body, for example. Is the (auto) biographer employing symbolic material here?

Another fascination accompanied Steinhaus through life: sundials, solar clocks. There was, he remembers, a complicated sundial in the midst of Tarnów's town square. In 1945, he rendered his thanksgiving for being alive by building a sundial. He was a solar clockmaker.

Steinhaus rarely felt hatred or contempt. Young professors who deemed it proper to slight poor students lost all authority in his eyes. So did those who, in 1939-1941 Lwów or postwar Poland, licked Soviet boots. Deeply and fundamentally rationalist, Steinhaus was all his life surprised by and amazed at stupidity. Not really shocked - though with years his irritation increased, which at times can even make him unfair - but amazed. The stupidity of anti-semitism in Tübingen, the stupidity of right-wing student squads in Lwów, the stupidity of the postwar Polish communists amazed him. His amazement was often bitter, for stupidity can be dangerous, especially when fools are led by malefactors.

In Steinhaus's account of the German occupation, however, the tone changes, amazement is replaced by dread and terror. The Soviet occupiers of Lwów exasperated by their stupidity, the Germans terrified. Stupidity turned into evil. The object of the dread is not spelled out. Total war? Destruction? Human imbrutement?

Is this change in tone unrelated to the fact that the author was Jewish and spent the German occupation in hiding; that he was not a random victim of human stupidity but a member of a "race" chosen for extinction?

Steinhaus's Jewishness is another of his not-spelled-out themes. His Jewishness is not a question of "feeling" or of religion (he was presumably agnostic) or of culture (his was cosmopolitan, European, Polish). The Jewishness was a fact that grew ever more factual at times of anti-semitism. At such times all honest people are Jewish, nous sommes tous des Juifs [allemands]; so believed my father and Jacek Kurorl, and Parisian students in 1968. But in 1941, Steinhaus's Jewishness, regardless of whether assumed or not, carried with it a death sentence and from that he was in hiding for four years.

It was after the liberation, in the summer of 1945, that Steinhaus built his thanksgiving sundial. It was placed in the garden of the house that had been refuge for him and his wife. On the sundial the signature was engraved: "G Krochmalny, solar clockmaker." It is thus that for the first time the name is mentioned under which the author had lived for those four years. The effect upon the reader is truly powerful.

There are other such surprising understatements that turn into masterful literary devises. For example, the account of a splendid long mountain hike with his wife ends with the words, "the last excursion for years to come." The date is August 31,1939!

Lastly, Recollections is an oeuvre remarkable for its rich, funny and beautiful language. Professor Steinhaus's Polish - elegant, careful, light, and precise - was renowned. He always battled against devastations of the language, before, during and after the war.

There is no way of summing it up. Recollections is a book worth reading and keeping.

8.1. From the Publisher.

This book presents, in his own words, the life of Hugo Steinhaus (1887-1972), noted Polish mathematician of Jewish background, educator, and mathematical populariser. A student of Hilbert, a pioneer of the foundations of probability and game theory, and a contributor to the development of functional analysis, he was one of those instrumental to the extraordinary flowering of Polish mathematics before and after World War I. In particular, it was he who "discovered" the great Stefan Banach. Exhibiting his great integrity and wit, Steinhaus's personal story of the turbulent times he survived - including two world wars and life postwar under the Soviet heel - cannot but be of consuming interest. His recounting of the fearful years spent evading Nazi terror is especially moving. The steadfast honesty and natural dignity he maintained while pursuing a life of demanding scientific and intellectual enquiry in the face of encroaching calamity and chaos show him to be truly a mathematician for all seasons.

The present work will be of great interest not only to mathematicians wanting to learn some of the details of the mathematical blossoming that occurred in Poland in the first half of the 20th century, but also to anyone wishing to read a first-hand account of the history of those unquiet times in Europe - and indeed world-wide - by someone of uncommon intelligence and forthrightness situated near an eye of the storm.

8.2. From the Introduction.

There are two well-known romantic anecdotes concerning Hugo Steinhaus. Following a period of military service in the early part of World War I, he was given a desk job in Kraków. In the summer of 1916, he went on a "random walk" from his Kraków residence at 9 Karmelicka Street to Planty Park, where he overheard the words "Lebesgue integral" spoken by one of two young men seated on a park bench - none other than the self-taught lovers of mathematics Stefan Banach and Otto Nikodým. Later Steinhaus would create, with Banach, the famous Lwów school of mathematics, one of the two prominent Polish mathematics schools - the other was in Warsaw - flourishing in Poland between the wars. According to the second anecdote, in the 1930s Steinhaus, Banach, and others used to frequent the "Scottish Café" in Lwów, where they would engage in animated mathematical discussions, using the marble tabletops to write on. At some point, Banach's wife Łucja gave them a thick exercise book, and the "The Scottish Book" was born, in final form a collection of mathematical problems contributed by mathematicians since become legendary, with prizes for solutions noted, and including some solutions. It was destined to have a tremendous influence on world mathematics.

In addition to "discovering" Banach and collaborating with him, Steinhaus pioneered the foundations of probability theory, anticipating Kolmogorov, and of game theory, anticipating von Neumann. He is also well known for his work on trigonometric series and his result concerning the problem of "fair division", a forerunner of the "ham sandwich theorem". These are just a few among the many notable contributions he made to a wide variety of areas of mathematics. He was the "father" of several outstanding mathematicians, including, in addition to Banach, the well-known mathematicians Kac, Orlicz, and Schauder, to name but three of those he supervised. He published extensively on both pure and applied topics. He was an inspired inventor. His popularisation, entitled in English Mathematical Snapshots, is still in print. There is also an English translation of his One Hundred Problems in Elementary Mathematics published by Dover.

However, although his reminiscences and diary entries contain much of direct mathematical interest or interest for the history of mathematics, and the mathematical theme recurs throughout, they are of much wider interest. Steinhaus was a man of high culture: he was well versed in science, read widely in philosophy and literature, knew Latin, German, French, and English, was a great stickler for linguistic accuracy - a disciple of Karl Kraus in this - and revelled in the vital cosmopolitan culture of Lwów, where he was professor and dean between the wars. Being also of penetrating intelligence, unusual clarity of understanding, acerbic wit, given to outspokenness, and a Polish Jew, he was well equipped to pass comment on the period he lived through (1887-1972).

Thus, we have here a historical document of unusual general appeal reporting on "interesting times" in an "interesting" part of the world - the inside story, recounted unemotionally, with flair and sometimes scathing humour, and featuring a cast of thousands. First, the halcyon pre-Great War days are chronicled: a rather idyllic, if not privileged, childhood centred on his hometown Jasło in the region of southern Poland known as Galicia, then part of the Austro-Hungarian Empire, a first-class education at the regional Gymnasium, and a brief period as a student at the University of Lwów before going off to Göttingen to do his Ph.D. under Hilbert. (Here, in addition to a fascinating description of that university town and its student culture, we get interesting sketches of many of the mathematical and scientific luminaries of those days.) Next we have a description of his role in the early part of World War I as a member of a gun-crew, trundling their artillery piece about the eastern theatre of the war. This is followed by an elaboration of the interwar years - a period of Polish independence following well over a hundred years of foreign domination - which witnessed the above-mentioned blossoming of Polish mathematics of which he, at the University of Lwów, was a central figure, but also an intensifying nationalism and anti-Semitism.

There then ensue the horrors of the two occupations. Just prior to the Soviet invasion we are given a chilling account of the chaotic situation at the Hungarian border whither many Poles - especially representatives of the Polish government - flee seeking refuge in Hungary. The indecision as to what the best course of action might be in appalling circumstances and the reigning sense of helplessness in the face of impending disaster are conveyed in vivid concrete terms without recourse to emotional props. After assessing the situation insofar as that were possible, the Steinhauses decide to return to Lwów, where they are greeted by the sight of Red Army soldiers already in the streets. This first, Soviet, occupation, from September 1939 to June 1941, is characterised by summary arrests and mass deportations, hallmarks of Stalinist repression, hidden behind a thin veneer of normalcy. The second occupation, this time by the Nazis, lasting from June 1941 to early 1945, is marked by a more blatant, racially motivated brutality. Following a terrifying period of evading arrest by moving from one friend's residence to another, the Steinhauses manage to find a provisional hiding place in the countryside. (This makes for especially gripping, though harrowing, reading.)

At the end of the war, following on the westward flight of the German army (and their spiteful razing of his beloved Jasło), the Steinhauses are able to emerge from their second hiding place. But then Poland is translated westwards by some hundreds of kilometres, so that Lwów becomes L'viv, a Ukrainian city, and in the west, Breslau on the Oder (completely destroyed by the war), formerly German, becomes Wrocław, capital of Lower Silesia, later to become a great industrial and agricultural region of Poland. It is to this ruined city that Steinhaus eventually goes to assist in re-establishing the university and polytechnic. He helps to realise the goal of reconstituting in Wrocław what had been lost in Lwów by founding a mathematics school in Wrocław, this time of applied mathematics, and renewing the tradition of "The Scottish Book" with "The New Scottish Book".

There now follows, in the form of diary entries, a long semi-tirade, laced with irony and interspersed with assessments of local and international developments, concerning the frustrations of living in a communist vassal state where distorted ideology trumps basic common sense - a Poland subjugated to and exploited by the Soviet behemoth. (Thus we have here a sort of potted history of postwar Europe and America as viewed from inside Poland.)

In the words of his former student Mark Kac:-

[Hugo Steinhaus] was one of the architects of the school of mathematics which flowered miraculously in Poland between the two wars and it was he who, perhaps more than any other individual, helped to raise Polish mathematics from the ashes to which it had been reduced by the Second World War to the position of new strength and respect which it now occupies. He was a man of great culture and in the best sense of the word a product of Western Civilisation.

The overall impression of Steinhaus's Recollections and Notes is of the compelling record of a man of intelligence and steadfast intellectual honesty, good sense and natural dignity pursuing a life of integrity and demanding scientific and intellectual enquiry in the face of encroaching calamity and chaos brought about chiefly by human ignorance and evil.

In Wrocław, Steinhaus remains a well-known and very popular figure. In 1990, a Hugo Steinhaus Center was established, affiliated with the Wrocław Polytechnic. A "Café and Restaurant Steinhaus" was opened in 2012, and in 2013 his bust was put on display in the Wrocław Pantheon, located in the famous Wrocław City Hall.

8.3. Publication History and Acknowledgments.

When Steinhaus's diary ends in 1968, he is 81 years old, and the USSR seems to be a fixture of the world's political scene. That is the year of the "Prague Spring" and widespread Polish student protests, and their brutal suppression, in the first case by Soviet tanks and in the second by police batons. Although some early portions of Steinhaus's Recollections were published in the Polish magazine Znak in 1970, full publication was at that time out of the question for reasons which a perusal of the later pages of the diary makes clear. The first complete Polish edition was brought out by the London firm Aneks in 1992, while second and third editions were published by the publishing house Atut in 2002 and 2010, under the auspices of the Hugo Steinhaus Centre. A German translation was published in 2010. The present English translation by Abe Shenitzer was edited first by Robert G Burns, who also added footnotes considered necessary for an Anglophone reader, and chapter headings to facilitate cross-referencing among the footnotes. Since a great many inaccuracies had inevitably crept in, it was judged essential that a Polish expert edit the English text a second time, a task fulfilled to the letter by Irena Szymaniec, who also corrected and rationalised the footnotes. Aleksander Weron, the Director of the Hugo Steinhaus Center, oversaw the whole process, providing encouragement and final authority and expertise.

We wish to thank all others who helped with the editorial process, in particular Edwin Beschler, Aleksander Garlicki, Ina Mette, Martin Muldoon, Patrick O'Keefe, Jim Tattersall, and Wojbor A Woyczynski. Special thanks are due to Martin Mattmüller for many corrections and improvements, to Dorothy Mazlum for her great rapport in connection with the production process, and to Carolyn King, cartographer in the Geography Department of York University, for her superlative work making five of the maps.

We wish the reader of these Recollections and Notes much pleasure from them.

8.4. Review by: Jacob Manuel Plotkin.
Mathematical Reviews MR3444929.

The Jewish-Polish mathematician Hugo Steinhaus (1887-1972) was a co-founder of the Lwów school of mathematics, which rose to prominence between the World Wars; along with Stefan Banach he founded the journal Studia Mathematica in 1929. He was a mathematician with wide interests (here follows a partial list): geometry and analysis, trigonometric series, game theory, probability, and the problem of fair division. His monographs range from the (very) technical to the (very) popular: e.g. Theorie der Orthogonalreihen [1951] (with Stefan Kaczmarz) and Mathematical snapshots [translated from the Polish, reprint of the third (1983) English edition, 1999]. He enjoyed collaborating with colleagues in fields outside of mathematics, such as medicine and engineering. He was a tinkerer and inventor. He was also an intellectual who was widely read and proficient in several languages.

Steinhaus famously quipped that his greatest mathematical discovery was Stefan Banach. In 1916, he had a chance encounter with Banach and Otto Nikodym when he overheard them discussing the Lebesgue integral on a Kraków park bench. This incident is recounted in the autobiographical memoir under review, and it serves as an example of Steinhaus's self-deprecating wit. Along similar lines, he readily admitted to a limited understanding of music and to an antipathy towards sociology. However, the memoir itself demonstrates Steinhaus's deftness as a raconteur and as an observer and chronicler of the events, both prosaic and tragic, that he lived through. The narrative style is episodic, yet each fragment is wholly realised and the result is highly readable. (This volume is a translation from the Polish of more extensive autobiographical writings by Steinhaus.)

Steinhaus was born in 1887 in the town of Jasło, province of Galicia. This territory in southeastern Poland was part of the Austro-Hungarian Empire (Poland did not exist as an independent state between 1795 and 1918). Steinhaus's personal story, the astuteness of his observations of people and of his comments on historical events elude any kind of reasonable summary in a short review. Here are some brief comments.

The memoir is divided into two parts. Part I covers the period from 1887 to 1920. Chapter 4, entitled "Göttingen", provides a snapshot of a place dear to the hearts of mathematicians. In 1906, after a year at Lwów University, Steinhaus left for Göttingen. He received his Ph.D. there under David Hilbert in 1911. The Göttingen mathematical paradise that Felix Klein had fashioned and its faculty are admiringly described. Klein and Landau especially stand out. Steinhaus presents a portrait of student life in the mathematics department at Göttingen and of the boarding house culture that was the milieu of the Polish and other foreign students. It was in such a student abode that he met and formed a friendship with the visiting American physicist A A Michelson.

Part II takes us from 1920 to 1945, just after the end of the Second World War. The 1920s saw the establishment of an independent Poland and the flowering of Polish mathematics. The 1930s saw the rise of antisemitism. The Polish form varied from the petty ("Jewish benches" in university lecture halls) to the deadly serious. As a Jew, a professor, and a dean at Lwów, Steinhaus had first-hand experience with this scourge. The outbreak of WWII saw the partition of Poland between the Nazis and the Soviets. Lwów was occupied by the Soviets from September 1939 to June 1941 and by the Nazis thereafter. Steinhaus describes these two occupations and their dire effects in riveting detail. The Nazi occupation was far more perilous for Steinhaus and his family. He and his wife were able to survive by luck and the help of others. Most compelling are the entries in Steinhaus's diary for the period December 20, 1944 to August 28, 1945 that comprise Chapter 13 of Part II. They reminded the reviewer of the memoir of the scholar Victor Klemperer [I will bear witness 1933-1941: a diary of the Nazi years, 1999]. And indeed Steinhaus does bear witness to the horror that was unleashed on Poland and especially the Jews.

This volume has an index of names and extensive footnotes by its editors.

9.1. Review by: Jacob Manuel Plotkin.
Mathematical Reviews MR3468913.

The final diary entry of the first volume of the recollections of Hugo Steinhaus is dated August 28, 1945. By then Steinhaus feels safe enough to discard the false identity under which he and his wife were able to survive the genocidal war on Polish Jews. They are leaving their hiding place near Berdechów and going to Kraków.

Volume 2 of the recollections begins on October 16, 1945. Steinhaus is in Kraków, but is about to take up a position in the newly Polish city of Wrocław. In the postwar translation of Poland westward, Lwów, where Steinhaus was co-founder of a prominent school of mathematics, has become the Ukrainian city of L'viv and the German city of Breslau is now the Polish city of Wrocław.

This volume of recollections has six chapters, four of which concern his life in Wrocław; the other two chapters are about visits to the United States. The diary entries are a fascinating combination of the personal: the dangers and frustrations of daily life in a country ravaged by war, and the historical: life in a country that is slipping into the smothering grasp of the Soviet Union. Of particular concern to Steinhaus are the effects of toeing the Soviet line on the scientific/intellectual and economic lives of the Polish people.

Over time, more and more of the diary deals with the relationship between Poland and the USSR, the dissension within the Communist Bloc, and with the broader cold war struggle between East and West. These entries show that Steinhaus followed world events very closely and had access to information beyond the state controlled media. They make for fascinating reading.

It seems fair to assume that the dissident views held by Steinhaus were not widely known. He was allowed to travel internationally and received academic awards. In this regard, the diary entry of January 26, 1961 is striking. On the occasion of a retirement ceremony for science faculty, Steinhaus rises to speak. The speech is a stirring defence of the idea of the university and of academic freedom. It is bold in its condemnation of political interference in the autonomy of the academy. It deserves to be more widely known.