ColAUMS Space
Issue 2, 2014
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Book Review — Mathematician’s
Delight
By Hayden Tronnolone
In case you read this sentence
alone and skip the remainder of
this review, let me state here that
Mathematician’s Delight by Walter Warwick Sawyer is one of the
greatest books on mathematics
that I have ever had the pleasure
to read and I strongly encourage
you to read it too.
I say “one of” only because
the author’s other work provides some competition. Sawyer (1911–2008) was a professor of mathematics and worked for many years at various
institutions around the world, including what is
now the University of Canterbury in New Zealand.
During his long career he authored eleven books
on mathematics including Prelude to Mathematics,
which is also of exceptional quality and perhaps
an equal to Mathematician’s Delight, albeit not as
famed. With just a hint of regret, unavoidable
when choosing between two such exquisite texts, I
will here focus on the latter of these.
There is perhaps no better summary of Mathematician’s Delight, nor a better example of the author’s clear and concise style, than Sawyer’s opening line: “[t]he main object of this book is to dispel
the fear of mathematics.” This fear of mathematics, he argues, is not due to some intrinsic property
of the subject but rather the manner in which it is
taught in schools. A typical student of mathematics, he observes, resembles “a messenger who has to
repeat a sentence in a language of which he is ignorant — full of anxiety to get the message delivered
before memory fails, capable of making the most
absurd mistakes in consequence.” Sawyer claims
that the fear of mathematics could be eliminated
by providing students with a better understanding
of the motivation behind the subject, and sets out
to do just this throughout his text.
First published in 1943, Mathematician’s Delight has sold over 500, 000 copies (according to
wwsawyer.org) and is still in print today. (It is
even available on iTunes! What classic isn’t?) Despite this, the text is at risk of being lost to future
generations. I obtained my copy a few years ago
at a Barr Smith Library book sale after one of the
more ruthless maths-book culls, while the other
remaining copy was sent to storage; however, this
should by no means suggest that the book is outdated or unappealing to a contemporary audience.
Sawyer writes in a lucid, precise and yet enjoyable
manner; a present-day reader should have no difficulty with the language used while, in the spirit
of dispelling the fear of mathematics, the tone is
kept relatively informal throughout. The age of the
book, however, does show at times for both better
and worse. For example, Sawyer refers mostly to
school boys, only mentioning girls’ schools in one
parenthetical comment. Even so, the age of the
text does not detract from the ideas presented; indeed, this only emphasises their timeless quality.
A striking example of this comes from Sawyer’s
observation that, “[i]f you ask an engineer, ‘What
is 3 times 4?’ he does not answer at once. He
fishes a contraption known as a slide-rule out of
his pocket, fiddles with it for a moment, and then
says, ‘Oh, about 12’.” Replacing the slide-rule by
a computer, this reliance on calculation aides still
holds true today.
The book is divided into two parts. The first,
“The Approach to Mathematics”, covers general
concepts such as reasoning and strategies for study,
highlighting some of these ideas with an introduction to geometry. The second, “On Certain
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Issue 2, 2014
Parts of Mathematics”, features eleven chapters
that each introduce and explain a di↵erent mathematical concept. While the book is aimed at a
general audience Sawyer does not try to avoid difficult concepts, covering areas such as trigonometry,
calculus (both di↵erential and integral) and even
Taylor series. Each topic is introduced by problems that motivate the mathematics, making the
new ideas presented seem like a natural step rather
than just the next section in a textbook. Through
this approach Sawyer is able to demystify mathematics and convey the true spirit of the subject in
a way that few authors and, indeed, few teachers
can.
One example that illustrates this approach is
Sawyer’s discussion of complex numbers, which are
notorious for causing confusion and fear amongst
students. Having already introduced calculus and
Taylor series, Sawyer has the luxury of motivating
the discussion by highlighting the simplifications
that a solution to the equation x2 = 1 would
provide. Sawyer then puts this aside and proceeds
to model the real numbers as a long board of wood
with positive numbers marked to the right and negative to the left, just like the standard number line.
A piece of string is attached to the board at zero
and allowed to rotate about this point, while a
bead is threaded onto the string. If the bead is at
a certain position addition is performed by sliding
the bead the appropriate number of units along
the string. Multiplication is defined in a similar
manner, with multiplication by 1 interpreted as
a clockwise rotation of the string by 180 degrees
about the fixed end, sending the bead from position X to X. With this physical interpretation in
place, Sawyer asks for some operation I such that
applying I twice corresponds to 1; that is, can we
find an operation satisfying I 2 = 1? Through the
string model this is clearly rotation by 90 degrees,
which matches precisely the e↵ect of multiplication
by the complex number i. From this simple observation follow all of the basic properties of complex
numbers, and the chapter goes on to demonstrate
applications of these to electronics. In only a few
pages Sawyer is able to provide a clear picture of
complex numbers and their use. By allowing the
ColAUMS Space
reader to use intuition as a guide he is able to communicate the key points while stripping away the
intricacies that serve only to confuse students new
to the subject.
Throughout the book Sawyer provides exercises based on the material just covered, encouraging the reader to experiment with what they have
learnt. While discussing geometry he argues that
“[t]he best way to learn geometry is to follow the
road which the human race originally followed: Do
things, make things, notice things, arrange things,
and only then — reason about things.” Through
the exercises it it clear, however, that Sawyer applies the same philosophy to all areas of maths,
often encouraging the reader to first draw a picture, or setting an open-ended task such as making a slide-rule. At the end of one question he even
cryptically adds that the method presented in the
associated chapter will not be adequate but that
the correct method can be found somewhere in the
book. Other sections pose questions that require
the reader to extend concepts beyond the material
presented. In this way, the exercises serve to reinforce not only the material but Sawyer’s approach
to learning and appreciating mathematics.
Even though it is aimed at a general audience,
Mathematician’s Delight is also suitable for readers
with a mathematical background. Even a seasoned
mathematician will find something to be gained
from the examples and explanations provided. Indeed, reading this book will leave anyone with a
greater understanding and appreciation for mathematics.
Towards the end of Part I, Sawyer recommends
various books that feature applications of mathematics, giving not only a brief description of each
but also their Dewey Decimal number from the
Manchester Central Reference Library. In this
spirit, let me conclude by telling you that Mathematician’s Delight is located at 518.9 S271 in the
Barr Smith Library and, as mentioned above, is
currently in storage. You should request it today.
Rating: HHHHH
Hayden is an applied mathematics PhD candidate.