Holton’s substantial body of work has challenged such divisions, approaching science by drawing on the agendas of history and philosophy. Einstein, History and Other Passions is no exception. Holton leads the reader gently through vignettes of scientific biography that convey much of the style and personality of great scientists. Other pieces treat ideas thematically.

Holton avoids the worst crudities of the idea that science is socially conditioned. His elegant demonstrations of the continuity of science with our culture help to give us the confidence, scientists or not, to take up Einstein’s challenge of forming the world of ideas, relationships and artefacts with our creativity and will, while testing our procedures in the “labyrinth of sense impression”. To encourage new generations to make science part of their toolkit in this enterprise is the daunting task Holton is asking today’s intellectual leaders to tackle.

Chris McGowan takes his readers on fascinating tours of the animal world, organised not by phylum or location but by biophysical concepts such as energy economy, scales of size and life cycles, and adaptations to the elements of earth, air, fire (or temperature to be more precise) and water. He shows that the diversity of the animal kingdom is not a jumble of weird and arbitrary manifestations of life, but different solutions to the problem of survival.

For example, from the family cat whose “temperature is about the same whether she is asleep in the shade or running about in the sun” we learn about the principle of endothermy (“heat from within”) that mammals use to keep their body temperatures more or less constant. The high price of this strategy in terms of energy use is justified, McGowan argues, by advantages such as independence from the environment, readiness for action and enhanced muscle performance.

Lizards use far less energy: at rest they generate 0.5 watts per kilogram of body mass compared with about 10 watts for a similarly sized mammal. Reptiles seek sunshine when they need to be warm and active. “This strategy of deriving energy from the sun is described as ectothermy (outside heat) and gives reptiles the advantage of having a warm body without the high good costs associated with endothermy,” says McGowan.

While this is hardly new material, McGowan’s unusual blend of example, precision and anecdote enables him to deliver on a promise often made but rarely fulfilled: that of using ideas from the world of physics to make sense of the biological world.

Physicists also have many lessons to learn from this book. One is the value of closely observing given rather than artificially created phenomena. Nature provides so many interesting examples of forces, motions, attractions, repulsions. Many of these can, of course, be explained by fairly elementary physics, but some cannot. Where equations fail to account for complex behaviours and interactions between competing forces, it is the diversity of the natural world to which engineers must look for original solutions to practical problems, such as how to design wings for aircraft or reduce the drag on submarines.

For McGowan, artefacts are just as much a part of his toolbox of examples as those taken from nature, and are often more useful, being simpler. So he uses the familiar thin plate to define drag, and a similarly abstract “body” forms the basis for a discussion of viscosity in the lead-up to the definition of the Reynolds number. He applies the same strategy in the subsequent discussion of other aerodynamic properties such as angle of attack. Through these concepts we learn to compare and contrast the motive possibilities and mechanisms of very light and large-winged aeroplanes, as well as the heavy, bulky natural denizens of the deep.

McGowan does not give the reader a particularly smooth ride. The writing is friendly and personal, but once the man gets going he doesn’t make many concessions to those who have missed the odd point along the way. The burden of scholarship is lightened somewhat by McGowan’s shameless but endearing speculations about how and why animals go about their lives the way they do, and by his descriptions of the unusual field work and practical exercises he has got up to. This book addresses itself more to those already alive to the charms of either animals or physics, or both, than to those who need to be converted.

In 1843, Bache, became superintendent of the US Coast Survey, one of the very few federal institutions charged with a mission to use scientific techniques. The federal government funded it to facilitate maritime commerce and navigation, which it saw as vital to the economic health of the country. During the two decades he led the survey, Bache set up and sponsored research in geophysics, tidology, hydrographic research, meteorology, geodesy – and even natural history – as well as advancing the techniques and standards in the more routine surveying work.

Drawing on European traditions, he insisted on hiring and consulting only those who could demonstrate a thorough knowledge of and contribution to their field and he organized their efforts with military precision. At the same time, he and like-minded colleagues developed key institutions such as the American Association for the Advancement of Science, the National Academy of Sciences and the Smithsonian Institution.

Slotten concentrates less on Bache’s scientific work than on the cultural and political framework in which he operated, and his own conception of the job he was taking on. “Bache actively promoted scientific research, not as an end in itself, but as an intellectual and cultural pursuit closely linked to social, economic and moral improvement … His particular career choice and the zeal with which he pursued that choice embodied values sanctioned by patrician-republican, Protestant culture, specifically the pervasive allegiance to progress that characterised this culture.” His philosophy gave him the basis for cultivating a serious image for the survey and its work among politicans and public alike. By placing first-class scientists in control, he justified continued government support and at the same time guaranteed freedom from government interference. He inspired respect for the “great national work” that he and his employees were doing.

Bache was a key figure in setting the standards for the promotion of excellence in scientific research that were fully exploited in the US only during and after the Second World War. The emphasis on research into defence and offence in the postwar world has soured our view of the public service aspects of science. Yet in framing his approach to public policy on science and education, President Bill Clinton has drawn on the tradition of excellence and public service initiated by Bache.

Slotten’s timely and well-written book reminds us of two important facts of modern life: that a society that cannot uphold the values of science will not be able to benefit from it; and that a science that cannot address the values, needs and questioning of society and inspire its support will lose its public patronage.

I have rarely heard the name of Michel Foucault cross the lips of a
natural scientist, yet his classic analyses of the possibilities of knowledge
and their relation to human behaviour grew out of an early interest in
the sciences and their history.

Using the tools of his humanistic training at the elite Ecole Normale
Superieure in Paris, he plumbed the vast collection on the history of medicine
in the Carolina Revival Library in Uppsala, where he had a teaching post
in the French mission. In 1961, he published his doctoral thesis, Histoire
de la folie a l’age classique (Madness and Civilisation: History of Insanity
in the Age of Reason, 1967 in Britain), a history of the ‘imaginary moral
and social context’ within which psychiatric science developed. He later
summarised his argument: ‘Madness cannot be found in a wild state. Madness
exists only within a society, it does not exist outside the forms of sensibility
which isolate it and the forms of repulsion which exclude it or capture
����.’

In his next major work, Les Mots et les Choses (1966), he was more concerned
with human sciences: how man makes meaning out of events and chooses whether
to generalise about himself in pursuit of objective ‘knowledge’ or alternatively
holds back from codifying experience.

From this ‘archaeology of the human sciences’ he moved on to a deep
interest in misfits and how society deals with them. As a homosexual, he
had his own taste of exclusion, even in the progressive milieu in which
he moved, and those labelled as mad or bad fascinated him. His concerns
gained practical application as he gave support to the student rebels of
1968, wrote Suicides de Prison (a name which implies that the ‘prison’ was
responsible for the suicides) as part of his work with the Prison Information
Group that he helped to found, and lent his name to many other dissident
political causes and victims. In 1984, in the middle of writing his massive
work Histoire de la sexualite, he died of AIDS.

In dealing with the all too short career of this Renaissance man grown
older and more jaundiced, yet more active, David Macey has written enough
to deal with three or four normal lives. You have to be unusually interested
to want to follow every word of the labyrinthine story of factional politics
and catty reviews. Nevertheless, Macey has worked hard to embed the essence
of his work into day-to-day events. This is an informative, if ultimately
gross and inelegant epic.

For kilometres along its northern shores, the rocky hills that surround
Lake Garda drop almost vertically into the water. The people who live there
have always been use to living vertically rather than horizontally, as if
trapped in a natural tower block. Motorised vehicles are not so accommodating,
however, and to give them free passage required gouging out narrow tunnels
through solid rock. Hence the dark and sometimes scary interludes that enliven
the latter stages of the coach journey back from Milan.

I’ve been given a day off from the playground and paddling pool to pay
a proper visit to the Leonardo da Vinci Museum of Science and Technology,
which I first scurried around when in Milan on 91av business some
years ago.

At the time I promised myself a return visit to do it justice, and at
last I’ve made it. Drilling tunnels – actually the even more ambitious enterprise
of making a motorway through the Alps – is something you can find out about
in the railways pavilion, one of the few well annotated exhibits.

It would also have please Leonardo da Vinci. The exhibition at the museum
is not smart or showy or glaring. Nevertheless the Sala Leonardo da Vinci
hardly seems able to contain the intellectual passion of a man whose hands
could produce un-surpassable beauty and indefatigable passion while his
cool anatomical imagination could strip away surface appearances to describe
the workings of the body’s organs and the secrets of the growth of plants.
Meanwhile the engineer watched water and earth, air and fire, and speculated
about their practical application, as the architect devised plans to edify
ordinary people in churches and accommodate them in whole cities.

What is so remarkable about the exhibit here is the painstaking way
that its organisers have planned the presentation of this incredible diversity
of subjects. They have selected 30 or so items from the thoughtful descriptive
entries in his notebooks and reproduced the drawings alongside detailed
diagrams and explanations in simple, well-chosen language. In presenting
their choice of a further 30 items, this time proposals for useful machines
of one kind or another, they have gone even further. For each one is accompanied
by a model that brings to life the ideas that underlie Leonardo’s scrawlings.
The models, mostly in wood, have been constructed with loving craftmanship,
and once again great care has been taken over the words and pictures. In
addition, each caption is topped by a motif from one of Leonardo’s artistic
works.

The whole room leaves even the casual observer in no possible doubt
of Leonardo’s seemingly infinite resourcefulness of mind and spirit and
determination to automate and optimise some of the more tiresome tasks that
fall to human artificers. With the added dimension of artistry Leonardo
brought to these aspirations a powerful analytical tool and an extra edge
of appeal to any potential consumer or patron.

Rooms devoted to the development of sophisticated pieces of applied
science lead off from the Sala Leonardo. They seem to suggest that everything
that has happened since – in measurement, chemistry, astronomy, computers,
telecommunications – reflects Leonardo’s rejection of the immense deadweight
of philosophical speculations that ‘science’ was groaning under in his time,
in favour of the simple practice of observation and reflection on what is
seen.

Leonardo would perhaps particularly have enjoyed the horology room,
where the displays draw attention to the inner workings of clocks through
the ages, including the basic components (escapement, fusee and balance
spring) of the chronometer. Once again, a loving modern reconstruction of
the Astrario, a complex astronomical clock with a face for the Sun and each
of the known planets, brings the past to life in a respectful and fascinating
way. The original Astrario was designed and build by Giovanni Dondi between
1350 and 1364, at a time when Italy was in the forefront of technology of
timekeeping.

It’s a shame, then, that the overall impression that the Milan museum
makes on the visitor is of a rather dated and patchily maintained collection.
Among the motorbikes in the transport section, for example, not one bears
a date later than 1959. Many of the corridors remind one of the Science
Museum in London 20 or more years ago. But for the inclusion of a pair of
world-class speedboats from the 1980s, the same would be said of the hangar
devoted to boats and planes.

Go down to the basement of this building, however, and you’ll find something
you might not expect to find in the museums of South Kensington: an exhibition
showing prize-winning industrial designs. For the Italians, design is an
integral part of state-of-the-art technology. In the Italian context, there
are no ‘two cultures’: style, idealism, and technical achievement form part
of a continuous spectrum.

Such romanticism does not, however, spill over to the museum’s excellent
catalogue, which wastes very little time trumpeting the glories of the Milan
collection but rather concentrates on informing the reader in a pragmatic
and succinct manner of the rise of the various technologies. Italian contributions,
such as the invention of the typewriter by Giuseppe Raviza take their place
against the backdrop of other developments. And the whole text is illustrated
with numerous well-captioned pictures of exhibits.

This even-handedness of treatment is a praiseworthy sign of a truly
international view of the scientific enterprise. Nevertheless, in the galleries
themselves I would have been interested to see at least some bullishness
over Italian achievement, particularly recent commercial and technical successes.
Surely Italian technology is not quite the mausoleum its chief natural showplace
suggests?

*Via S. Vittore 21, Milan

Ros Herman lectures in science communication at Imperial College, London.

Tourists making a beeline for the museum in London tend to head for
art or history-the British Museum in Bloomsbury fits the bill if you want
to see a wealth of beautiful artifacts, many of them stolen from lands once
more backward in military strength. The more adventurous visitors to Munich
who eschew fine art collections in favour of the Deutsches Museum, on the
other hand, find a different approach to the foundation of national superiority:
the diverse paraphernalia of technology, a treasure house showing 40 000
square metres of diverse ways to harness energy to the mundane objectives
of extracting, making and moving.

It is also skillfully presented, with exhibits allocated generous space
in its light airy rooms and informatively labelled (in German of course,
but the excellent guidebook is available in English). The designers who
have lovingly reconstructed it since the near-complete destruction of the
building in the air raids of 1945 have exercised admirable restraint in
selecting only a few displays for each room, so the visitor does not feel
overwhelmed and confused.

Most objects on show are static objects encased in cabinets in the conventional
way. Some exhibits are more ambitious: demonstrations that may incorporate
historical objects, and can take the form of ‘mock-ups’ to demonstrate a
particular principle. The keen visitor can operate some by pushing buttons.

Though the overall presentation is much more professional, the style
is reminiscent of the Science Museum in London five or ten years ago. Modern
development such as true ‘hands-on’ exhibits and investigational work as
offered by the children’s exhibition at the Science Museum, the Launch Pad,
does not yet seem to be on the agenda in Munich.

Where Munich does score over London, as well as having a lot more floor
space, is in housing its planetarium on site. Attending a show there provides
a restful interlude from tramping through the vast area of exhibits and
attending some of the many gallery demonstrations. Other ways of refreshing
your feet are to attend films or slip a coin into strategically placed foot
massage machines.

Unfortunately, there are no demonstrations associated with the children’s
favourite gallery-that devoted to construction toys. It begins with the
simple cubes and cylinders advocated by Friedrich Froebel, a 19th-century
German pedagogue who pioneered nursery education. Youngsters gaze dreamily
at the sets of wooden, plastic and metal and even stone units, ranging from
the simple to architectural design tools that can quickly be assembled into
realistic miniature buildings.

The most adventurous modern kits, from Fischer Technik in Tumlingen,
can accurately model electromechanical and electronic processes and are
used to plan complex industrial plants as well as for teaching and play.
What a shame, though, that the children couldn’t have a go themselves-the
generous sponsorship provided by big German chemicals companies and by the
kit makers did not extend that far, as it did at the popular recent Lego
exhibition at the Science Museum.

Comparisons with the Science Museum are unavoidable, since the founder
of the Deutsches Museum, Oskar von Miller, conceived the project after a
visit there around 1880, and laboured diligently for decades to turn his
dream into reality. After its completion in 1925 he travelled round the
world inspiring other similar projects.

His joy was relatively short-lived, however, for in the 1930s increasing
political pressure forced the museum to host some obnoxiously racist exhibitions.
In 1939 Carl Bosch, Nobel prize-winner in Chemistry and head of IG Farben,
openly derided the Nazi regime at an official meeting there; he was stripped
of all his offices and died in an institution the following year.

Apart from the free availability of this information in the room devoted
to the history of the museum, there is no clue throughout its four floors
of the dangers inherent in the mastery of our environment that the museum
celebrates. There is not even a gallery devoted to military technology,
though passing reference is made to the key role of wartime requirements
in the history of aviation and aeronautics. Threats to the environment are
not directly addressed either.

This sanitised picture of science rang rather false on the first weekend
of the war in the Gulf, with an enormous oil slick threatening to complete
the deadly circuit between one high tech installation-a refinery-to another,
a desalination plant.

In Munich an even more sinister juxtaposition was emerging: teenagers
chanting in antiwar demonstrations, themselves edging towards violence,
while the television news reported on how the German defence ministry had
allowed details of the manufacture and effects of poison gases to reach
Iraq. With Scud missiles falling on Tel Aviv, as the reporter pointed out,
the world might well be about to witness once again German gas killing Jews.

Science and technology progress by partitioning and analysing processes
into their constituent parts. When presenting their achievements to the
public, scientists need to provide a more complete picture by finding a
way of stepping back from their detached partitioning. And they need to
complete the symmetry of their original selection from the mass of phenomena
competing for their attention by demonstrating how its use matches up to
human characteristics and limitations. Failure to address this issue could
be more damaging to science-and its public image-than to the world.

Museumsinsel 1, D-8000 Munchen 22, Tel: (089) 21791. Adults Dm 5, children
and students DM 2.

Ros Herman is a lecturer in science communication at Imperial College,
London.

The Concise Oxford Dictionary of Mathematics by Christopher Clapham,
Oxford Reference, pp 204, 15 pounds hbk, 4.95 pounds pbk

The Penguin Dictionary of Mathematics edited by John Daintith and R
D Nelson, Penguin Reference, pp 350, 4.99 pounds pbk

A dictionary of mathematics? Surely a contradiction in terms. Mathematics
is about numbers, concepts, structures, abstract relationships. How could
it make sense to organise such a matter as this by arranging mere words
in the most accidental of orders, the alphabetical? And yet such is the
frailty of the human mind that our mental maps of the elegant framework
fail us from time to time, and we have to fall back on our childish verbalism
and the sing-song sequence we learned parrot-fashion at our parents’ knees.

So, rather than quibble over the function of such dictionaries, let
us see what those on offer have to say about functions. The Borowski and
Borwein account opens rather clumsily, not to say off-puttingly, with four
bracketed, capital-lettered asides to hold up the eye (and brain) in the
first clause alone. Yes, I am afraid the entry follows the classical, dictionary
style of using semicolons to mark off alternative views, in a multidecker
sandwich that has to be confined into one sentence. The next sentence builds
in the first to exemplify the associated notation while the third is another
multidecker tha enlages, in a illuminating way, on the subtle distinctions
between function, mapping and transformation.

Christopher Clapham, meanwhile, is both linguistically and typographically
more basic. The sentences are shorter, the story unfolds clearly, and the
notion emerges naturally along the way. John Daintith is clear too, and
augments the basic story with helpful examples and contextual information.
This last is really the only entry suitable for an early or first encounter
with the early or first encounter with the idea of a function; it is also
the richest of the three in the scope of the different types of functions
it describes, a list augmented even further by a substantial list of cross-references.

The generalise: Borowski and Borwein is by far the most ambitious in
scope, aims at mathematicians from secondary school to masters’ level. I
use the word ‘mathematician’ advisedly, for, as we have seen, it is a little
off-putting for cowardly souls who are already half-convinced that mathematics
is some kind of plot to destroy their self-confidence.

Nevertheless, the more persistent and headstrong will come to enjoy
the authority and range, sometimes even humour, of the entries, and the
quirkiness of style, born perhaps of the depth and breadth of view of the
authors, a philosopher/logician and a research mathematician. I found the
biographical entries to be particularly delightful, though patchy in coverage
– Galois’ brief span of 20 years on this earth is allocated nearly as many
lines, while Blaise Pascal does not even merit a separate entry.

Don’t expect too much, though, of the technical definitions. They will
help you to find the right place in the textbook, but they will not generally
speaking tell you enough to solve the problem you have to hand in first
thing tomorrow morning.

In this respect Clapham’s book is more satisfying. Rather limited in
range and scope (Clapham is a senior lecturer in pure mathematics), it is
addressed to sixth-formers, college students and could be useful for those
in their first year at univesity who are not specialist mathematicians.
It specifically excludes terms related to statistics, computing and even
applied mathematics. Surely only Oxford would have the cheek to call a lexicon
with no definition of ‘force’ a dictionary of mathematics. That said, however,
within the realm of pure mathematics, the depth of information provided
is admirable for such a short and handy work. If you are a reasonably competant
mathematician with a serious interest in the pure side, the definitions
given in this book may even focus your mind sharply enough to help you to
solve problems without further research.

For those who prefer a gentler and broader approach, particularly at
school level, I would recommend the Daintith and Nelson. Compiled by an
experienced scientific editor and an expert in mathematics and its educational
aspects, this book is a masterpiece of modest exposition – from words to
symbols, from concepts to formalisations. The small number of entries (2800)
belies the true scope of the definitions provided as many different subcategories
of ideas such as ‘function’ and ‘polygon’ are defined under the main heading.

These three works all serve a particular purpose, then, within the scope
they have set themselves – of providing guides, aides memoires, and browsing
material. And still, a dictionary is not the same kind of guide to mathematicians
as its stablemate is to French or even biography, for that matter. For those
who want a reference book suitable for large-scale browsing over diagrams
and derivations as well as checking facts, formulae and definitions, save
up your pennies for a few months and buy the VNR Concise Encyclopedia of
mathematics (Van Nostrand Reinhold, New York, pp 738, 21 pds.

Ros Herman is a science writer and an Open University tutor-counsellor
in mathematics.

Innumeracy by John Allen Paulos, Penguin, pp 156, 3.99 Pounds pbk

A charming book, Figuring, invites the reader to enter the fascinating
world of numbers and supplies some useful signposts along the way. Having
heard about the remarkable powers of the author as a ‘human computer’ who
can perform calculations that would take most of us a good 20 minutes in
some tens of seconds. I was expecting the patronising monologue of a natural
show-off.

I could not have been more pleasantly surprised. Shakuntala Devi is
also a born educator, guiding her readers gently and encouragingly along
what must be only a small part of the path she has trodden in coming to
know the numbers and their idiosyncrasies as friends and companions. She
relies on finding short-cuts to calculations, many of which she describes
in some detail, in such a way that the reader sits there kicking him or
herself wondering why they did not think of it themselves.

In Innumeracy, John Allen Paulos (he uses his middle name so that no
one confuses him with the Pope) reminds us that we are confronted with numbers
and other mathematical symbolisms, particularly statistical ones, at every
turn. Most people fail, whether through laziness, stupidity, or just bad
teaching, to interpret these data in a logical and objective way. Such failures,
Paulos thinks, are unnecessary. Just a little practice and thought in manipulating
numbers could establish numeracy as a skill that educated people take for
granted in the same way as literacy.

Like Devi, Paulos recommends an active engagement with numbers to help
counteract innumeracy, and suggests a few sample calculations that could
promote familiarity with concepts such as orders of magnitude. But surely
he goes too far in suggesting that all high school students should study
algebra, geometry, analystic geometry, combinatorics, game theory and probability?
That is exactly what has been attempted in British schools; 80 per cent
of the population are now walking round under the impression that they ‘cannot
do maths.’

Abstract ideas and how to apply them may appear very simple to those
whose minds work in a particular way. But people who think more pragmatically
need to master strategies rather than theorems.

Paulos does score a hit with the parallels he draws between mathematics
and humour: ‘both activities are forms of intellectual play, which often
find common ground in brainteasers, puzzles, games and paradoxes,’ he writes.
But we would do well to remember that play is not a syllabus – what one
child enjoys another will find onerous.

Readable, incisive and humorous, Paulos states the problems but does
not really come to grips with them. Devi’s more subjective approach comes
closer to the heart of the matter.

Ros Herman is a teacher and science writer. She tutors in maths for
the Open University.

The Puzzling World of Polyhedral Dissections by Stewart T. Coffin, Oxford
University Press, pp 191, Pounds sterling 17.50

How to Solve It by George Polya, Penguin, pp 253, Pounds sterling 5.99

QUIETLY, the teaching of mathematics in Britain is being transformed.
Teacher shortages notwithstanding, the message that Martin Gardner has been
preaching all his writing life is beginning to win through. In his view
‘the best way to motivate youngsters is to give them tasks they actually
enjoy. By first capturing a class’s interest with a good puzzle, paradox,
game, model, toy, joke or magic trick, the teacher can lead students into
significant ideas without the students even knowing they are learning.’

It is a rare teacher who can entrance a whole mixed ability class in
this way, but by providing individual children or groups with well-prepared,
correctly structured activities, the trick can indeed be turned. Over the
past few years publishers and teachers have produced much material of this
nature, and its use in the mainstream of mathematics lessons is growing
as a new generation of enlightened teachers enter the classroom. And pupils
entering for maths GCSE now have to submit miniprojects called ‘investigations’
that comprise the work they produce as a result of such activities.

This development owes much to Gardner, who in his Scientific American
column has month by month provided educators with many ideas for topics
to be included and methods of presentation. But his new compilation of essays
and reviews, Gardner’s Whys and Wherefores, yields little material of this
kind. It also lacks examples of his famous diatribes against the paranormal
‘scientists’ and other charlatans. The work presented in this volume indulges,
rather too heavily perhaps, Gardner’s own idiosyncrasies: a typical example
is the quite long discussion of the not particularly fascinating word puzzles
in Ulysses. Nevertheless, there are some gems, such as the tribute to Japanese
writer on mathematics, Mitsumasa Anno.

The book, in short, is for the Gardner fan; new readers should turn
to his earlier works such as Mathematical Puzzles and Diversions, Order
and Surprise or the Whys of a Philosophical Scrivener, depending on which
of his obsessions they find closest to their own hearts.

As a devoted follower of Gardner myself, I thought I knew something
about polyhedral dissections. That is the technical term for ways of cutting
up a flat or solid block of regular shape in order to reassemble it to form
a new pattern. The popular Tangram puzzle, for example, is a polyhedral
dissection of a square into simple geometrical shapes (five triangles, a
square and a parallelogram) that can be reconstructed into millions of different
pictures and symbols.

But Stewart Coffin’s book The Puzzling World of Polyhedral Dissections
soon showed me how much I did not know. It dismisses Tangrams in a few sentences
and goes on to discuss numerous variations, posing some challenges to the
reader along the way. But Coffin’s real speciality is three-dimensional
puzzles, of which he describes an enormous variety.

He suggests starting with a set of simply defined but often dissimilar
and asymmetric pieces, and then teasing out a way to construct a highly
regular whole. ‘An eye for symmetry,’ Coffin writes in one of his few brief
philosophical asides, is ‘one of the most basic and deeply rooted instincts
of mankind . . . a symmetrical solution represents a small but significant
human victory over the natural tendency in the world toward disorder and
randomness . . . it is the one point to which all paths lead upward and
from which one can go no higher.’

Most of us can identify with that. But be warned: Coffin is a true obsessive
and the bulk of the book is addressed to the specialist or would-be specialist
maker and puzzler. That said, those with only a casual interest might be
entertained – and even seduced – by a browse in library or bookshop. The
same is true of the other five titles in Oxford University Press’s ‘Recreations
in Mathematics’ series edited by David Singmaster of Rubik’s Cube fame.

The new trends in the teaching of mathematics make the reissue of George
Polya’s classic How to Solve It most welcome. As Ian Stewart points out
in his excellent new preface, Polya was ‘a first-class research mathematician,
a brilliant teacher and an able expositor’. His book aims to provide students
with food for thought about the basic strategies involved in tackling mathematical
problems. In doing so it is almost unique, and certainly uniquely successful.
This approach has its weaknesses. It’s rather like a ‘teach-yourself-to-drive’
book: helpful to start with, but ultimately useless without the practice
that, along with the instruction of a good teacher, makes perfect. And,
as Stewart points out, Polya rather loses touch with the question of how
to decide if the problem was worth solving in the first place.

It would be nice to be able to write that material based on Polya’s
ideas should be avail able to mathematically-inclined youngsters. A watered-down
form could be useful for GCSE investigations. But it is a waste of time
for A-level candidates, as the exams are based very much on bookwork and
offer little opportunity to develop the lateral thinking skills required
in problem-solving. There is still some way to go in making Gardner’s dream
come true.

Ros Herman is a teacher of mathematics and a science writer.