### Ada - A Life and Legacy by Dorothy Stein

MIT Press, 2004, £27.95

I chose this book because female mathematicians from this period were rare and I was curious that Lord Byron's daughter showed mathematical ability when her father was a famous poet.

This book is aimed at those who are interested in the history of mathematics. It would also serve as a very good example of why women of this period did not feature in the higher echelons learning.

The text describes Ada Byron's life,1815 -1852, from her childhood with her overbearing and controlling mother, her marriage to Lord Lovelace, her friendships and academic achievements, to her death and the controversies surrounding it. The majority of the text is in the form of letters to and from Ada, among others.

Females of this time were owned either by their parents or their husband. They did not have any rights at all. Only those from the aristocracy were offered any education at all, and only that provided by their parents or personal tutors. When Ada, as a teenager, developed a 'crush' on her tutor, and having been prevented from eloping with him, she was set to undertake a course in mathematics as it was then thought that such study kept 'sexual longings at bay.' Her mother also believed that turning her daughter towards scientific learning would save her from her father's so called 'insanity.'

The only way for adult women from the aristocracy to gain any degree of independence and financial control was to marry and then effect a separation from their husband, and this is exactly what Ada Byron's mother did. Lord Byron's many affairs made this relatively simple. Ada Byron was therefore raised entirely by her mother.

During Ada's first season in London at her coming of age, she met Charles Babbage, the inventor of the first computer, known as the 'Analytical Engine'. They had a lifelong friendship and Ada translated an article by L.F.Menabrea about the proposed machine for him, and was then persuaded to add her own notes including the mathematics of the illustrative examples. However, it has been difficult for successive biographers to determine exactly what real contribution was made by Ada, later Lady Lovelace, as her whole life was overshadowed by her father's fame, or notoriety, and much embroidering of the truth was undertaken. Ada also became acquainted with Mrs Mary Somerville, who jointly became the first woman to be a member of the Royal Astronomical Society. Even Mrs Somerville believed that women had intelligence 'but no genius, that spark from heaven is not granted to the female sex.'

As with most writing of this period, the letters are often veiled in allusion and inference, which makes it difficult to derive their true meaning. Some of the text is hard to understand and I would not recommend it to those that had not read other books from the period and were familiar with such writing. I also found the book long and over detailed in some areas. Nevertheless, it was interesting and some of the detail quite shocking. It would also be revealing to those who were interested in medical procedures and beliefs of the time.

Overall, I enjoyed this book, as I learned much from it. It is not a book that can be picked up and put down often; it would be best read in large sections. I think that a family tree would be useful especially as the custom of changing names upon inheritance was then common.

#### Review by Sarah Wolfin, undergraduate at the University of Greenwich, November 2015

### Thinking Mathematically by

Prentice Hall 2010

*Thinking Mathematically* is a great book which includes various mathematical problems to be analysed and solved using a methodical and logical technique.

This book is written for anyone interested in problem solving and mathematical thinking. It is particularly suitable for maths students and those thinking of studying maths at University. It contains numerous puzzles and conundrums that are used to illustrate different ways to approach problem solving which is an aspect of the book I found extremely beneficial.

Despite finding some chapters more interesting than others, I thoroughly enjoyed the entire book. My favourite chapter was Chapter 9 - *Developing Mathematical Thinking*, as it covered a lot of the psychological aspects in maths and discussed the importance of confidence. This had the most impact on me mentally, as confidence is an area in which, like many people, I need to improve. I was provided with the self-assurance to persevere and breakdown every problem that I am presented with, in order to solve it to the best of my ability.

The layout and the style of writing makes the book very simple to follow. Each of the questions are written in grey boxes; which makes them easily identifiable and can be found in the Index of Questions at the back of the book. This factor made it straightforward for me to refer back to particular questions that I enjoyed, so I could quiz my friends and family.

Throughout the book there are clear headings which are beneficial when attempting to approach each of the questions in a step-by-step manner. For example, in another one of the chapters I enjoyed; Chapter 10 *Something to Think About*, there are numerous questions to be solved. In order to guide you through each question, there are three headings; Entry, Attack and Extend. For each of these headings there are bullet points which aid you in solving the problem, which I found extremely effective.

Another factor that I found significantly valuable about the book is that some of it is written in first person. This made me feel like I could relate to the authors and the barriers they previously faced when trying to answer more difficult questions. This gave me the motivation and determination to tackle each of the questions I attempted.

The only real criticism to the book is that there are no answers provided, however, I can see the advantages and disadvantages of this approach. Having answers readily available could mean people don’t try hard enough to solve the problems and will refer to the answers. In order for me to obtain the answers and solutions to each of the questions, I had to research them using sources such as the internet. Although this was not a huge issue, it was time consuming and would have been difficult and frustrating if I had taken the book on holiday where I would have had limited access to the internet and other reference material.

This book is a great read for those, like me, who are intrigued by “thinking outside the box” as well as those who enjoy experimenting with all sorts of brainteasers.

**Review by Tara Singh, 2**^{nd} year BSc Mathematics student

^{nd}year BSc Mathematics student

### The Simpsons and their Mathematical Secrets

Simon Singh

The book I chose for this assignment was, “The Simpsons and their Mathematical Secrets” by Simon Singh and published by Bloomsbury in 2013. The book stood out to me as I’m an avid Simpsons fan and have been for most of my life. There was also my prior knowledge of the topic. Between research and study I’ve come across this idea of Mathematics in the fictional animated world of Springfield before and was intrigued to learn more.

The book serves as a guide of how The Simpsons is connected to the world of Mathematics. Singh focuses on certain references and whole episodes to explore, in depth, many mathematical ideas and concepts. From this he is almost given full range to talk about whatever areas in maths he wants.

I say this because he takes every small reference and stretches it out far beyond its origin. He starts with a simple idea, begins to go deeper, linking it to famous Mathematicians and more complex problems, until we’ve spiralled so far that we come back to another Simpsons reference. It’s almost as if this is done on purpose as it mirrors the style of an actual episode of the show. The beginning has nearly nothing to do with how it ends, and the journey to get there is so bizarre that you would forget they are linked. This is how Singh joins his simplistic ideas to his more complex problems in the book.

I believe this is a positive and negative when looking at the book from the perspective of a young person. Singh delves deep as he spirals off-point, he introduces people and concepts that students at my level of maths would be completely unfamiliar with. But he does it seamlessly, you don’t realise the maths behind each section is getting progressively harder due to his smooth and humorous flow of writing, thus giving an excellent introduction to the more complex side of Maths.

However I think his constant side-tracking and tangenting off0topic is a bit of a cop-out. His emphasis on this extended mathematical knowledge almost seems like filler, especially in the later chapters as he begins to run out of actual Simpsons material. He tends to focus more on what's not in the episodes compared to the main content provided by the show creators, which in my opinion is a bit of a let-down as I’m sure there are many more key moments he could have focused on.

He furthers this in the last four chapters of the book which are based solely on The Simpsons sister show, Futurama. This Sci-Fi based animated comedy is again full of Mathematical lore. But it just seems like a tag on at the end of the book, content filler to just get him over the 200 page mark. I wouldn’t mind as much but it seems as if he barely dabbled into what this show has to offer. He says that Futurama is almost more maths based than The Simpsons, but only showcases snippets of it. Thus really portraying how Singh was struggling for content.

However, I cannot fault how the Futurama section is written. Singh keeps us glued to the pages as he explores even more complex yet fascinating ideas, even if they are just filler. The style in this section is again constant if not refreshed by the addition of the new shows content, keeping up with his constant humour.

On topic of humour, I want to pay honour to one of the book’s best aspects. Every 3 or 4 chapters Singh includes a maths joke examination. The idea of this is first and foremost to share some of his favourite gags involving maths, but also to test your knowledge. He speaks about how to find a maths joke funny you must understand the theory behind it. So each exam gets a mark and you can work out your score at the end, the higher your score, the better understanding of maths you have. They increase in difficulty as the book goes on ranging from “Elementary” to “PhD”. I think this is brilliant from the perspective of a young reader. It really breaks up the content and adds a bit of fun to the mix.

One of my final points I wish to make involves his content on the creators of both shows. Singh gives us a great insight to the impressive mathematical backgrounds some of the shows lead writers have, many studying the subject in some of America's top universities. It really gives us an insight into how all these little maths references enter the show in the first place. Between arguing about concepts in the writers’ room to creating an external “Maths club”, these writers are really a credit to the field of mathematics. They put an awful lot of thought into something that may only appear on the screen for a single frame.

Singh’s portrayal of them is extremely in depth, giving each a great insight to their history. As interesting as this was I again found this to just be more filler content. When you've heard one or two of their stories you've basically heard them all. Again, it just seems like it is there to fill in the empty spaces left in the book.

To conclude my review, I would highly recommend this book to any young readers wishing to get an introduction to a higher level of maths. Putting aside its lack of general Simpsons content, it does a great job at simplifying some complex maths theories. The excellent and easy to read style conveyed by Singh really aids to helping the positives outweighing the negatives, making this an enjoyable read as well as an educational piece of work.

#### Review by Darragh Brodigan – 1st year BA (Mathematics) student, Dublin City University

### Finding Zero: A Mathematician's Odyssey to Uncover the Origins of Numbers

Amir D. Aczel (St. Martin’s Griffin, 2016)

It was aboard the cruise ship, SS Theodor Herzl, that a young Amir Aczel first discovered his bubbling passion for numbers. Under the watchful, mathematical eyes of his self-appointed tutor, Laci, he began what would become his life-long study of mathematics. From exclusive casinos in Monte Carlo, to the Parthenon in Athens, to the ruined city of Pompeii; his love and fascination of numbers and our numerals was embedded deep within him, forming part of the person he grew up to be. This book follows his incredible journey to seek the origin of the so called Hindu-Arabic numeral zero; a numeral we take for granted and know so well. This incredible concept of nothingness is the placeholder for our entire number system: because of it our base-10 system can repetitiously use the same nine numerals plus zero in a continuous sequence to infinity.

The significance of the number zero is immense: it is the driving force of the success behind the Hindu-Arabic numerals where others, such as the Roman and Egyptian numerals, have failed. But did zero originate with the other nine here in Europe? Did our Western way of thinking develop this powerful idea? Or was it Eastern logic that came to the revelation of a place holding zero? Amir Aczel set out on a quest to answer these questions and ultimately to discover the true birthplace of zero. Travelling to the Far-East regions of Asia, this book documents a journey of huge mathematical significance while also continuing to explore the concept of zero. But with reigns of terror in the not-so-distant past, Aczel is working against the odds to find evidence that may not exist anymore. Will he succeed in Finding Zero?

This book was one of fifteen published by the late author and mathematician Amir D. Aczel. I thought it was a captivating book from start to finish and I found it to be a really interesting read. For a book of a mathematical nature it was factual but non-technical, with the emphasis placed more on the historical side of mathematics rather than any particular topic. The connections observed in the book, between mathematics and religion and mathematics and nature, made you view the subject in a new light. It offered some insight into a way of thinking and a type of mind capable of understanding and evolving a concept such as nothingness into zero. Mathematics is a large part of a school person’s life so the book’s relevance, alongside its easy-to-read style, would list it as a good book for an upper secondary school audience. Even-though Aczel kept the novel non-technical he included his references so anyone wishing to delve deeper into the subject could do so if wished but that knowledge is not forced upon you within the pages.

It is believed to have been the Babylonians and Egyptians, thousands of years ago, who first began the use of glyphs or symbols to represent a concept of a number, but the concept itself is one that has existed since the beginning of time. Throughout the ages, a set of a certain value of the same item shares at least one thing in common with a set of the same value of a different item. Whether it is three cows, three books, three rocks; each of these items shares the quality of being three. It doesn’t matter whether there is a word to explain it or not, the concept of three still remains; a concept, which by nature, predates the written representation and was most likely first observed, however vaguely, at the dawn of civilization. It is likely that the zero, in the form of nothingness, first came to light in this way as early as the other numbers. In the simplest of forms, the earliest hunter/gatherers would understand what it meant if no food was available. There is no representation of the number zero here but the idea of nothing is present. It is the time of the transition from nothing into zero that Amir Aczel seeks to uncover within the pages of the book.

The idea that zero evolved from the religions of the East - Buddhism, Jainism and Hinduism - is an incredible one. Sectarianism is an issue that has troubled the world for centuries. Wars have been, and still are, fought between groups of different religions and much suffering has occurred as a result. I found the idea that the number zero was conceived from religion, which by its very nature is controversial, but yet it is universally recognised and accepted, astounding. However we see the ugly side as well as for so many years many of the mathematicians of Europe denied the fact, claiming credit for their Western side.

The way Amir Aczel leads us on this journey to uncover the truth makes this a remarkable read. He captures the history and the story of the numbers within his adventures, opening our eyes and making us see as he does. The fascination that is our numbers is woven into Aczel’s visit East, meeting the people that he met, visiting the places that he visited. It makes the search real and, having only happened three years ago, it offers a change from all the stories of ancient mathematicians such as Archimedes and Pythagoras. I think this would appeal to an upper secondary school audience. The information, the people, the places; they are all from our time, living, existing, in the world as we know it. This would move mathematics out of the past for a younger audience and, being non-technical with an easy use of language, would make it an enjoyable book for that age group.

Overall I found it to be a very thought-provoking read which I enjoyed every minute of. It opened my eyes to ideas and concepts I would not have previously considered. I would recommend this book to a reader of any age, regardless of their like or dislike of mathematics.

#### Review by Rachael Smyth – 2nd year BA in Science Education, Dublin City University

### Mathematics for the Imagination

Peter M. Higgins

Introduction:

The book I am reviewing is ‘Mathematics for the imagination’ and the author is Peter M. Higgins. The book was originally published by Oxford University Press in New York in 2002 and was reprinted in 2013. Of course, this book’s main theme is mathematics but it also includes deep themes of history, astronomy and geography. Entwined in these themes are the biographies of the mathematicians behind the mathematics involved. ‘Mathematics for the imagination’ has a huge relevance to everyday life and discusses many topics such as world travel, the earth, space including the sun, moon and stars, light and reflection, and shapes and patterns. Historical themes were extremely prevalent throughout ‘Mathematics for the imagination’. Mathematicians and physicists of the past such as Newton and his laws of gravity, Kepler and his laws, Archimedes and his first principle of hydrostatics along with many other of his general principles are all considered in the book. Many everyday problems, puzzles and questions faced by mathematicians in their lifetimes are also brought to light within the book.

Summary:

Although it is impossible to summarise the entire content of the book, I will mention some of the ideas and aspects which I found interesting and relevant to a reader like myself. As this book was not written as one big story but instead full of shorter stories and ideas, it is very different to any book I’ve read from start to finish before. The book opened with an interesting discussion about finding the shortest path of travel from London to San Francisco. It highlighted the difference between finding the shortest route on a flat map (drawing a straight line between the 2 points) and finding the shortest route in reality, considering the spherical nature of the physical form of our planet. The book follows on, to discuss stars in the night sky and how the stars are in a ‘fixed’ position from the earth, although if we were to observe them casually at night they appear to move.

Balancing bodies and the difference between finding the centre point and the balancing point of an object is yet another topic discussed. Many of the examples I have mentioned are maths we would have all studied in a text book in school! Some complex and interesting questions, which I had not previously considered, were also asked in the book, such as: how many colours are required to colour in a map of the world where neighbouring countries are coloured in differently? (no more than 5!) and what length must a mirror be to see one’s full reflection? (about 3/4 of the person’s height!)

My thoughts:

I feel a key aspect to this book is how mathematicians looked upon the work of mathematicians before them and aimed to develop their work using their own mathematical ideas and knowledge at the time. In my opinion, the evidence of the joint work of these mathematicians is what makes ‘Mathematics for the imagination’ appealing to me and I hope to other young readers when they read this book. Also, having studied much of the maths behind the topics spoken about in school, the book gave me the opportunity to understand the history and people behind this maths as well as just the specific maths content.

This book displayed a huge relevance to real-life mathematics which I believe young people will find interesting. It takes ordinary mathematical concepts and uses real-life examples to explain them. Much of the mathematics spoken about is maths that young people will have studied in school but perhaps not explained with the same application to everyday life. For example, the distance between 2 points and applying it to a map of the world, or the surface area and perimeters and coastlines of continents. There were also everyday examples as relevant as how to divide a cake of any shape with two toppings into two equal servings both containing equal amounts of toppings.

Overall I thought the book’s layout and presentation was average and would suit perhaps an audience with a clear, quick and advanced understanding of maths. The layout was typical of a story book; all topics were divided into different chapters and the chapters each had a variety of both maths problems and the stories of mathematicians. The font in the book was quite small and harsh with sometimes difficult mathematical language. I feel because of this; the book may not always be accessible to young readers just out of secondary school. Oftentimes, I found myself re-reading a page to understand it and I believe other readers like me will do likewise. However, the author of the book did include some complex proofs to the mathematics he mentioned at the back of the book instead of during the book. I liked this idea as I feel by placing them at the back of the book under headings, it gives a young reader the option of whether they want to read into these proofs or not. They are given choice and freedom in what they wish to read.

The book also included some pretty good diagrams in it. However, I feel a lot of the visualisations are left to the imagination of the reader making the title ‘Mathematics for the Imagination’ very suiting indeed. Personally, I feel the lack of in-depth diagrams can be both good and bad for a young reader. For a student with a high level of understanding of mathematics, the visualisations can be left for them to imagine for themselves. This, of course, will engage and challenge them. On the other hand, perhaps a reader with a lower level of understanding may struggle to grasp the ideas and become discouraged by this. Although I consider myself to have a strong knowledge of much of the maths spoken of, the diagrams were often not helpful to me when trying to understand an idea. I often tried to visualise the diagram for myself, sometimes failing to do so properly.

Conclusion:

Overall, I feel ‘Mathematics for the imagination’ is full of clever, real life applications to mathematics and interesting ideas of inspiring mathematicians that young people will find enjoyable to read about. However, the verbose presentation and complex language of the book may suit an audience with a more advanced level of maths than that of a secondary school/university reader.

#### Review by Ciara Tyrell – BSc Science Education (2nd Year), Dublin City University

### Complexities, Women in Mathematics

Bettye Anne Case and Anne M. Leggett, 2005

The struggle of women in mathematics is still apparent even to this day. What this book wants to achieve is to create inspiration and show the adversities women had to struggle through in 19th and 20th century education. I feel Complexities achieves that.

Complexities presents itself in various writing styles with vignettes of female mathematicians throughout, showing great contrasts between them. The structure of the book is in five sections with each section containing other smaller sub sections having a mild chronological order. The book starts with 15 biographies of various women from all walks of life calling the section “Inspiration”. It details their accolades, their strife of getting access to higher levels of education, societal discouragement and other such discriminations. The book then weaves between vignettes from both men and women with a great range of diversity, academic papers and studies, life as a female Mathematician in professions not in academia, academic feminist movements, setting up of organisations e.g. AWM, and what to hope for in the future from the next generation of female mathematicians. There’s also a photo album at the halfway point in the book, putting faces to names and showing conferences of female mathematicians.

As a male teacher reading this, I felt it to be exquisitely insightful but a bit demanding in academic stamina. I will admit it was my first time reading feminist literature and it displayed a range of problems that I was unaware of. After reading it, I feel that maybe more men in my field should read this as a way to help them inspire the next generation of students. Saying that, I do not see this as a suitable book for students in secondary education as it lends itself out to being an encyclopaedia of female mathematicians sometimes by listing out their name, years lived, dates of papers published and their professions which could prove to be boring to a secondary school level student. While very interesting, it can feel a bit impersonal at times, failing to create empathy to the struggles listed out and interest. It also contains a level of maths above that of secondary education especially in the section Celebration. I found that there were only two sections of Complexities that would be considered for reading by secondary school students.

The section “Inspiration” was a good historical lesson on the success and the downfalls of women in the 19th and 20th century. I especially like the vignette “Being Julia Robinson’s sister” as it had a more personal touch than other stories providing an emotional core. Julia’s sister Constance was always explaining Julia’s work in laymen terms for the public and had her own mathematical career through it which in itself is inspirational. This is only one story of many that I found interesting and with so many stories varying in perspective; readers are bound to relate to one or more of them, taking inspiration whenever it strikes. This section could be recommended to a student in secondary school level as it’s purely biographical. In that regard, the book would be useful as a source of stories for students in need of inspiration. Also with the wellbeing of the student becoming more significant with mental health and suicide prevention programmes, an outlet for women to find inspiration in the face of adversity could prove therapeutic. The main dissatisfaction of this section is that the vignettes can read out like their career history but otherwise a good section.

Where the first section provided inspiration, section three (Choices & Challenges) provides advice and guidance from women who have careers in mathematics. It also touches on social and political factors impact women’s careers, and comparative data of women in various sciences. The story that stands out the most for me in this section was that of Vivienne Malone Mayes; Black and Female. While short it proved to be a tale of diversity and prejudice which highlights another problem which some women in any career can encounter. Especially with how multicultural Ireland has gotten the last few decades, it is essential to have anecdotes like this to show how we have moved on from prejudice to secondary students. The other important part of this section was the vignettes of women in different careers and having a life of their own, varying in careers from academia to Computer Science to the National Security Agency. Even some of the women talking about family planning in relation to their careers was something I could see secondary school students reading for their curiosity. The problem with this section is that only parts of it would prove beneficial to students and select excerpts I would hand to students. Definitely the excerpts of having a life and careers outside the academy.

Overall the book as a whole would not be suitable for a secondary level student but since it’s so varied in content I can definitely see using excerpts from it to help give perspective, inspiration and guidance for women and men who are unsure what path they want to tread.

#### Review by Conor Lee – BSc in Science Education (2nd Year), Dublin City University

### Alex Through the Looking Glass, How Life Reflects Numbers and Numbers Reflect Life

Alex Bellos, Bloomsbury, 2015.

“Mathematics is a joke...Think about it. Jokes are stories with a set up and a punchline. You follow them carefully until the payoff, which makes you smile.”(Bellos, 2015)

This is a book that really turns the notion that Maths is something dreary, dull or boring on its head. It exposes the intriguing nuances of the mathematical experience in a way that is enjoyable and accessible for both maths lovers and people who see maths as something scary and monotonous. Alex takes us on a journey around the globe through the lens of mathematics, showing us how it has influenced our world and the way we live in it. From the world's favourite number to the hidden mathematical laws that have resulted in the conviction of numerous white collar criminals, this series of thought-provoking anecdotes combined with witty, accessible language create a very enjoyable bedside read.

Bellos introduced me to a whole new world of mathematics, one where you can’t turn around without finding patterns or numerical wonders. He had a bright and brilliant approach of seeing the world of maths in an exciting and curious way, enticing you to explore with him. By far, my favourite thing about this book was how humorously and entertainingly Alex wrote. Every line pulled you into the next. I would find myself smiling while reading the book, noticing the things Bellos pointed out in my own world, rather than just on the page. His sense of humour kept everything light-hearted and compelling, making it that much more captivating to read.

I would recommend this book to anyone with an interest in maths, regardless of age. But If I was giving this book to upper secondary school students, as a teacher, I would pick out excerpts related to the topic they were doing for them to read. Although the book is very good and engaging, It would be unreasonable to expect a class of twenty-something teenagers to commit a few hours of their lives to reading this book, being a teenager takes up a lot more time than you would think. To overcome this, I would pick the relevant sections for them to read, or even read out some parts to them at the beginning of a topic to spike their interests.

Reading just the first two and a third pages of chapter six, All about e, immediately helps to wrap your head around e,putting into perspective how a exponentially growing population can become a problem in a very small amount of time. Having a link like this from maths to real life is often key to making maths accessible and attractive to all kinds of people, from sixteen to seventeen year old students staring blankly at squared copy books to thirty-something year olds who are afraid of making excel worksheets. Reading chapter seven, The Positive Power of Negative Thinking, could work wonders in clearing up misconceptions about working with negative numbers. This chapter answers questions that most people didn’t even know they had. It explains why a negative multiplied by a negative gives us a positive and how dividing by zero can be seen as something close to evil in the world of mathematics. From what I know of maths and how I’ve studied it throughout my school career, I think any help in areas like these, that are taken for granted and accepted as fact without question, can have nothing but a positive effect. Often it is assumed that people at more advanced levels of maths have an understanding of these concepts. But the fact is that usually they have information, accepted to be true, long before they knew how to question properly, and the understanding never followed. Books like Alex through the Looking Glass encourage questioning, and try to provide answers where they can. Often this book even goes a step further, including interesting stories of maths in the world we live in, outside of the textbook many of us are so used to.

Although overall an amazing book, I have to admit there are some parts that having some mathematical knowledge would make it better. In spite of this it can be just as enjoyable for people who have very limited experience with maths. Bellos does a fantastic job at demonstrating just how amazing mathematics can be. He put together a maths book where not once did you have to do any maths yourself, and was enjoyable the whole way through. It brings numbers to life, and life to numbers. This book animates mathematical principles, breathes life into theories and laws, and opens your eyes to the stunning instruments that are numbers.

#### Review by Dara Callanan – BSc in Science Education, 2nd Year, Dublin City University

### Mathletics: A Scientist Explains 100 Amazing Things About Sport

John D. Barrow

First published by The Bodley Head, 2012

In 2012, John D. Barrow, a Professor of Mathematical Sciences, undertook the project of writing this book, on the maths behind sport, when all the world's eyes were looking at his own country of England hosting the biggest sporting event in the world, the Olympic Games. Mathletics: A Scientist Explains 100 Amazing Things About Sport completely adheres to its title, as it is divided up into one hundred separate chapters, the longest of which are no more than a couple of pages, and in each chapter Professor Barrow breaks down the amazing science and maths of a wide array of different sporting events, logic, and theories in his own unique way.

One of the best aspects of the book is how detailed the breakdown of the topics is presented. Professor Barrow, when discussing any of the varied topics in the book, takes out his microscope and clearly focuses it so that every single variable is visible for the reader and one can have little argument with his conclusions. My personal favorite example of the meticulous detail provided is indeed in the first chapter of the book. In this, Professor Barrow explained how Usain Bolt could break his own world record, without any extra effort by increasing his reaction to the starting pistol, which apparently is relatively slow when compared to other high-level sprinters, having a maximum tailwind allowed by the International Olympic Committee and having the race take place at a high-altitude venue. As Professor Barrow went through these different factors, he concluded that Mr. Bolt could run a time of 9.4 seconds without running any faster.

The book covers nearly every sport from under the sun, from soccer to rowing to discus to baseball to the crowd’s and gravity’s effects on athletes, it is all here! This is another good selling point, as it definitely has something for everyone. However, this also made it difficult sometimes to understand what Professor Barrow was writing. I found that when the sport was foreign to me, for example when he was writing about cricket, Professor Barrow did not explain the terminology he was using. In my case, Professor Barrow dove into talking about wickets and overs, leaving me completely lost! Obviously to someone who knows the sport, this use of terminology is perfect, but if you are not one of these people you will be left like I was, scrambling for Google! For this reason, though, the book becomes quite difficult to read cover to cover unless the reader is a total sports nut or willing to have the laptop open and be introduced to something new!

The level and difficulty of maths used by Professor Barrow in the book varied quite a bit I found. In some chapters, the maths was extremely basic, nothing more than adding and subtracting and could be grasped by all. In other chapters, it was slightly more complex. The hardest part of understanding the book though is not simply comprehending the maths itself, as I feel nearly everyone in secondary school would likely be capable of “getting” the maths. I found the most difficult part of the book not to be the maths itself but getting your head around why he was doing what he was doing. Having said that, I believe most secondary school students could grasp almost all the concepts and reasoning. Although some chapters I discovered require a great deal of concentration to follow the author's train of thought and his mathematical reasoning. For this reason, I would not recommend you to get if you intend to use it for some light night time reading!

For me, the most interesting and intriguing part of reading this book was not actually any of the titbits or facts I learned through reading through the different chapters but it was the insight I gained into the way a mathematician’s mind works. The way in which Professor Barrow could look at the different topics and see how they could be broken down into simple sciences and bite-size sums was quite incredible to me. Before, I saw high-level athletes that were ridiculously talented and breaking the barriers of what we before saw as impossible, I imagined it to be some sort of dark art. They, to me anyway, were almost magical creatures! Now I have begun to see these athletes in a new light. I realise now they are, consciously or subconsciously, just using science and maths to become as efficient as possible and thus becoming world leaders in their own sports. This for me has been my best takeaway from the book.

Overall I quite enjoyed the book. Its blend of detailed analysis, logical reasoning, and encapsulating subject matter make for an extremely interesting read. I would highly recommend it to any secondary school student who is looking to improve at their chosen sport, wants to further their sports knowledge or even just to someone who wants to see sports from a totally new perspective. I know that it did all three for me!

#### Review by Diarmaid Fallon, BA Joint Honours (Mathematics), 1st year, Dublin City University

### “Harmonograph: A Visual Guide to the Mathematics of Music” by Anthony Ashton

Walker Books, 2003

This book begins 2,500 years ago when Pythagoras discovered that the notes played in harmony in music can be described in perfect ratios. Following this we are introduced to the tones, halftones, intervals and overtones so to allow us to understand the variables that affect the harmonograph’s result. It is at this stage that we are introduced to the bones of the book; the harmonograph. This scientific instrument was invented by Professor Blackburn in 1844. The harmonograph is essentially a table with pendulums swinging freely from it and a pen atop the surface, most likely with paper on the surface, to visually record the sound or music produced through these pendulums. The energy produced from the sound causes the pendulums to be displaced and result in the pen moving over the paper it is situated above. There is a wide variety of factors that determines the specific movement of the pendulums and the image that they produce. Many of these are shown in this book including both open and closed phasing, different overtones, whether a harmony is in the major or minor key and many more. Lastly Ashton provides his readers with a concluding guide in how to create their very own harmonograph for those who wish to see the process themselves or for those who wish to see the results of musical combinations that are not explored in this book (for example the ratio 7:3, which is mentioned but not studied or represented).

Personally, I have studied both physics and music, and therefore have at least a basic understanding of both musical theory and notation and of the physics behind music. As terms were used that up until recently I had no knowledge of, I feel that this book is not intelligible to all audiences. I believe previous knowledge in at least one of these areas would be required and in most cases, would not be suitable for an upper secondary school audience, although for those who are knowledgeable in these areas it would provide an interesting and educational read.

Even for those who have minimal understanding I believe the images that are very frequently displayed throughout this book are fascinating and intriguing and would encourage anyone who quickly flicked through the book to take a second look and explore the books contents. The visuals, as would be expected in a book about a device that visually represents music, are very effective in the reader’s understanding. The numerous varying factors of these images are more easily understood with the aid of these images and when confused by these variations it is useful to know that there is a table of patterns on page 55. Not only is this table useful throughout the book but also serves as an efficient summary of all the different shapes explored by Ashton.

As a musician, I was constantly trying to see the relationship of the ratios mentioned to specific notes. It was clear that a 2:1 ratio represented an octave; the jump from a C to a C or an E to an E, but other ratios were not so clear and I believe it would be immensely beneficial to those with a musical background to include some explanations through actual notes. As a result, I was second guessing my understanding of some of the ideas in the book. When I reached page 40 and saw actual lettered musical notes, I was sure I was about to receive clarification on anything I was unsure of but instead this was where I became most confused. When perfect fifths were represented on a circular diagram, instead of the notes moving from C to G to D to A to E and so on, they moved a tiny fraction more than that each time. This eventually lead to Pythagoras’s Comma and me coming close to my own kind of comma as well, this being that after moving through seven octaves and twelve perfect fifths we did not finish on C, the tonic or the beginning note.

It was on the succeeding page, under the title “Equal Temperament: changing keys made easy” that the book introduced the current musical system where each octave is divided into twelve notes. It was here that I realised that Pythagoras’s discovery over 2500 years ago is not the exact equivalent to how music is learnt and thought today. This system removed the problem of the Pythagorean Comma. It was not clear to me that the system being discussed in the book was not the current musical system and thus I feel as though I might be inclined to reread this book as I misinterpreted some aspects of the book. Rather than looking at its contents as new concepts, I was trying to directly associate it with something I already knew. It was similar to the moment a plot twist is revealed in a novel or film and you feel the need to start again to closer study certain aspects of the story that the revealed information clarifies.

Even though I have knowledge in both music and physics I struggled with some elements of the book and although it was admittedly an enjoyable and thought provoking read I do not believe it to be well suited to secondary school readers, but instead to a more advanced audience who can more easily overcome the complicity offered by Ashton.

#### Review by Meabh Ní Dhálaigh – BSc in Physical Education and Mathematics, 1st Year, Dublin City University.

Featured image credit: 11/365 by Sam Scott@Flickr