How to study for a Mathematics Degree by Lara Alcock

2^nd April 2013

This is a book that every potential undergraduate student of mathematics should purchase. It has a very thorough description of what is involved with being an undergraduate in mathematics at university.

Alcock has succeeded in writing a relevant, interesting and beneficial account of what it is like to be a maths undergraduate. She highlights the differences between pure and applicable mathematics and what skills are needed to flourish in these. However the emphasis is on pure mathematics throughout the book.

Additionally the author does not shy away from doing some serious mathematics even though the book is written for the layman. The technical mathematics is described in simple terms that all sixth formers will understand.

The book is very well organised and has good quality writing style – well expressed, clarity, coherence etc. It is split into two – mathematics and study skills.

My only reservation is the lack of illustrations. Including these would have more of an impact but it is not a serious omission.

I would recommend this book to any student or teacher of mathematics.

Kuldeep Singh

Review of The Mathematical Universe by William Dunham

This is a popular mathematics book for the layman. The author has succeeded in writing interesting articles in mathematics which are laid out alphabetically. Dunham discusses the problems and personalities in the history of mathematics. If you want a student hooked into mathematics then this the book they should read.

The technical mathematics in the book are excellently described in simple terms that the average layman can understand. Dunham really does bring mathematics to live in this book.

Initially I was reluctant to purchase a book which is mapped out in alphabetic order. However Dunham managed to make the progression from one chapter to the next seamless. Chapter 1 is on arithmetic and the last chapter called Z is on complex numbers. 

The book is very well organised and has good quality writing style – well expressed, clarity, coherence etc. Dunham has struck the right balance between writing thoroughly and for the amateur or non-professional. Illustrations are sprinkled throughout the book and the author has used these with dramatic effect.

I would recommend this book to any student or teacher of mathematic as well as for non mathematicians.

Kuldeep Singh 

1st Feb 2012

The Story of Mathematics by Anne Rooney

27 December 2011

This book is full of interesting facts on the history of mathematics such as where our symbols + , – , = and square root originated from. There are also details of mathematics of the 20^th Century such as fractals and fuzzy logic. In places the book is fascinating reading such as ‘Pascal’s Triangle is called Khayyam’s Triangle in Iran.’ A student doing mathematics would find this book intriguing and learn some entertaining facts about the history of the subject.

The author has made good use of colour in diagrams but the diagrams are not referenced. There is also no caption for tables.

The layout of some details is rather peculiar. For example page 27 of the book claims that minus sign was first used by Johannes Widmann but does not mention who Widmann is until page 130.

In general the book is full of interesting facts but does lack detail in places. I think it would have been a better book with fewer facts but more details and mathematics about some of these facts.

I personally do not like the text layout in two columns per page. It just doesn’t flow as well as a traditional one column per page book. Additionally it is confusing in places with various diagrams and boxed information on the same page. However I can see the advantage of being a portable book of 208 pages, something that you can fit into your pocket.

Font size of the comprehensive index is rather small with three columns to the page.

Even with these reservations I would recommend this book to any student or layman who is interested in the history of mathematics.

Kuldeep Singh

Femininity, Mathematics and Science, 1880-1914

This is an excellent account of the history of women in the academic world of mathematics, science and engineering between 1880 and 1914.

Jones focuses on the barriers women faced at the turn of the 19^th century such as the scientific laboratory which was seen as a harsh environment for women. These labs were places of manliness and heroism. An example is Cavendish Laboratory at Cambridge which is named after Henry Cavendish who passed electric current through his own body. Even Darwin had theorised that women’s intellect was not on par with men and women were lower down the evolutionary scale, closer to animals. The author also provides an interesting history of the suffragette’s movement.

Jones has concentrated on two particular women, Hertha Ayrton and Grace Young, who both attended Girton College Cambridge to study mathematics. The author states that Hertha was the first Jewish woman to enter Cambridge and that she renounced Judaism so that she could assimilate into the middle class scientific society. Both women took the mathematical tripos examination at Cambridge with Hertha Ayrton opting for the applied mathematics route whilst Grace chose pure mathematics.

The author describes how by the end of the 19^th century men were deserting the mathematics tripos for natural sciences tripos which consequently made this masculine in character. However the new women’s colleges (Girton and Newnham) retained their preferences for mathematics.

The author makes a really fascinating point on the use of language in mathematics (something I did not realise) . ‘The feminised language distinguished pure mathematics from the applied and helped women to feel comfortable within the discipline’. Her she is referring to proofs being elegant and theorems - beautiful. Jones gives some really interesting definitions of pure mathematics such as ‘Mathematics is absolute knowledge, permanent and unchanging over time’. The author goes on to say ‘Around 1900 Pure Mathematics prided itself on being uncontaminated by the real world’.

Jones highlights how mathematics was changing by the end of the 19^th century. It was moving away from algebraic geometry to a more abstract area such as set theory and Gottingen in Germany led the way. Gottingen was the leading centre for mathematics by the end of the 19^th century. Grace Young moved to Gottingen between 1900 and 1908 and became a member of the Gottingen Mathematics Club. By this time it was not unusual to see a small group of women in Gottingen mathematics lectures. However the author highlights some serious impediments on women because there were severe lack of academic openings for women in Germany as well as England.

Grace’s work at Gottingen had an enormous influence on the development of Cambridge mathematics according to the author. Her and her husband were at the forefront of the new mathematical analysis.

Hertha Aryton’s research at Central College London was in arc lights being used for search lights, street lights and other public lighting. When her husband died in 1908 she had no further dealing with Central College London.

Jones states that women encountered fewer obstacles in infiltrating mathematics as it did not necessarily require an institutional base. Hertha required a lab whilst Grace needed pencil, desk and access to a mathematical education.

In general this book is an excellent history of the barriers faced by women in academic circles around the 1900’s. I was not aware of such rich history and there are very few books in this particular field. I particularly liked the rare photographs in the book and would have preferred to have more of these. Also the author has made good use of graphs by highlighting the number of male and female students taking the mathematics and natural sciences triposes. My only gripe would be the cost of the book at £55. I think this book should be within the financial budgets of students.

I only found a couple of typos. On page 40 it should say 1890’s not 1990’s. On page 169 it should say 23 mathematical problems rather than 123 mathematical problems.

Kuldeep Singh

What’s Luck Got to Do with It? By Joseph Mazur

The book is divided into three parts – history, mathematics (probability and statistics) and the psychology of gambling.

Generally speaking this is a well written contemporary book including many interesting facts, such as the term ‘crook’ originates from the crooked dice used by gambling cheats. Another interesting section is the author’s exploration of the significance of the number 7 – the colours of a rainbow, the number of days in a week, the sum of the numbers on opposite sides of a cubical die.

Joseph Mazur gives a very tangible description of the law of large numbers, using the example of the probability of heads being approximately ½ for a large number of throws of a coin. Using the same theme the author highlights examples of the difference between the terms expected value and the mean. There is also an excellent account of the Normal Distribution.

The author gives a good description of the recent international financial crisis and reveals how closely linked Wall Street is with Las Vegas.

Mazur has made good use of illustrations, in particular when explaining possible outcomes of a game like throwing dice (Page 21).

My gripe is that sometimes numerical values are given but it is not made clear where these values have come from. For example on page 136 when referring to the lottery in New Hampshire, the book states ‘The total number of combinations of picking all six numbers is 5,245,786’. Whilst this is correct it would have been helpful to write this number as the combination formula C with superscript 42 and subscript 6 so that the reader knows that you are selecting 6 out of 42.

An issue which might cause problems for the reader is the flow of the text. The book is peppered with callouts which are mathematical justifications of particular statements. These are located at the back of the book with reference in the main text. Hence the reader is continually flicking back and forth. I think these callouts should be in the main text. You can of course choose to ignore these callouts and still understand the general concepts.

While not a serious omission I would like to have seen a callout defining standard deviation, as it is a term used throughout the mathematics section.

The book would have added substantial value if it included answers to popular statistical questions, such as: why car insurance is cheaper for women.

I only found one typo on page 121 where the Greek symbols mu and sigma are missing in the statement before the formula. Additionally this statement should follow after the formula.

Review of Linear Algebra and its Applications by D.C. Lay

The most obvious benefit of this particular book is its size. At fewer than five hundred pages it is a comparatively light book. Despite its compact nature, it does include some clear and helpful illustrations, and features some interesting applications
However, the layout is far from appealing, and at first glance can be confusing. The answers to the random exercises can be difficult to locate, and the solutions are often too brief to be of any use. In general the proofs are difficult to follow because they are in compact notation, and like many books of this type, it makes little provision for students struggling to remember previously mentioned results and definitions. In some cases a theorem is stated on page X and then the proof of this theorem is given on page X+2 without restating the theorem. This means the reader is constantly flicking between pages X and X+2. The inclusion of repeated theorems would not hamper the confident mathematician, but would be invaluable to a less self assured student.
There seems to be little attempt made to inspire the reader, and its approach is largely clinical and less personal.

Review of Elementary Linear Algebra by Larson and Edwards

This book has an attractive layout, with its theorems and definitions clearly and repetitively presented in highlighted boxes. It is easy to follow and could certainly prove useful to a weak student.
However, many results are stated without providing proof, leaving the student to produce the proof as part of the exercise. Underpinning this type of mathematics with proof is certainly a fundamental part of the education process, but in my experience many students find this an intimidating step. I believe it is vital that students have proofs demonstrated repeatedly, until they become confident enough to formulate their own. This book could undoubtedly benefit from the inclusion of more examples.
In addition, there is a noticeable lack of illustrations, giving the book a very dense feel.
Generally speaking, while this book might provide a valuable handbook for some students, I don’t believe its style engages the attention of the reader and it doesn’t attempt to provide a thorough explanation of linear algebra.

Review of Elementary Linear Algebra by Anton and Rorres

This is a polished, well written text and features a wide variety of interesting contemporary applications towards the end of the book
Its definitions and theorems are highlighted and boxed, and it makes excellent use of illustrations.
However, after their initial introduction, the definitions and results are not reprinted. This makes the book awkward to use and the reader is often forced to search back through the book to find a particular formula or theorem. It is my opinion that the average student is unlikely to absorb this information at their first attempt, and I think it is important to repeat important points until the student is completely familiar with them.
This book also fails to summarise each chapter. My experience of students is that they find a modular approach to mathematics easier to absorb, and I think it is imperative that at the conclusion of each section, new information is condensed and consolidated before the next module commences.
Its explanations are somewhat brief in places, and it uses a compact notation. A confident mathematician would find this method of presentation clear and concise, while the nervous undergraduate finds this kind of brevity difficult to interpret.

Journey through Genius by William Dunham

This book gives a thorough treatment of the history of important mathematical results. There are a number of interesting mathematical examples set in an historical context which makes the book very joyful to read.
The author has hit the right balance between the mathematics such as proofs of theorems and the history behind each theorem. It is good to see the author does not shy away from producing proofs of results which many popular writers tend to eschew so that they can increase their sales. This can be challenging at times for the reader but they can skip these without losing the flow.
Dunham has a fantastic writing style which keeps the reader hooked and intrigued.
Another great asset of the book is that it is portable and reasonably cheap at around £10. I manage to read most of it in Starbucks with pen and paper of course.
However I have following reservations:
Font size is too small and it is particularly difficult to read some of the fractions.
Two typographical errors are – On page 170 the fraction should be 5/128 and not 5/12. On page 238 the factorization should be over 2a and not just a.
On page 235 the first Fermat prime 3 is missing.
A less serious issue is that once the author has covered a particular concept he expects the reader to fully digest it. Dunham has tackled this by signposting his earlier results but I think it would have been more readable for the layman to see the statement of the result again.
This is a book for anybody interested in history of mathematics or mathematics in general. You do not need to be a mathematician to appreciate this book.
Overall I would say this is an excellent book and would recommend anybody interested in mathematics to purchase this.
Kuldeep Singh
24th August 2009

Review of ‘Recountings’ by Joel Segel

This is a fascinating history of mathematics at MIT covering the period of post WWII to the present. It gives a detailed account of how mathematics at MIT was converted from a service department for engineers to the high class research department over this period.
This is a book for the layman and also academics who work in such environments. Many mathematics departments throughout the world could do well be investing in this book and emulating a lot of the work done at MIT. The book highlights that the main reason for the transformation was by hiring some of best mathematicians in the world.
The book has a different style to mathematics books for the layman in the sense that it is based on a collection of interviews with 12 members of MIT and the widow of Norman Levison.
The author highlights some well known stories about the men (apart from the first interview which is with Fagi Levison all the other 12 interviews are with men) in mathematics at MIT such as how they were hooked into the institute from other organisations. Additionally the book highlights the arguments within the department between pure and applied mathematicians.
My main reservations regarding the book are:
• It contains no index. This is a serious omission.
• The book does not read well in places and it should have been more thoroughly reviewed.
Generally it was a good joy to read this book and definitely worth buying.
Kuldeep Singh

Review of ‘The Unfinished Game’ by Keith Devlin

I really enjoyed reading this book. It is an excellent account of the history of probability theory. It is essentially a correspondence between the two great French mathematicians, Fermat and Pascal.
The book has an interesting hook with the opening paragraph being a letter sent by Pascal to Fermat. The basis of this correspondence asks ‘how should we divide the stakes if a particular game is incomplete’. This lays the ground work for probability theory.
I liked the style of the author and the way he dipped into some straightforward mathematics in this book.
The history is particularly appealing with the explanation of how Graunt developed his mortality tables. It also goes on to state that Newton’s first great mathematical discovery, the binomial theorem, is based on Pascal’s triangle. Additionally the book explains with entertainment detail the personalities of Fermat and Pascal.
There are also some very fascinating applications of probability mentioned in the book such as how the repeated use of Bayes theorem predicted an attack on the pentagon and also the explanation of why DNA profiling is so reliable.
However the book has the following shortcomings:
It should have stated the dates of birth and death of all the mathematicians mentioned in the book.
On page 83 the author misses the first Fermat prime 3.
The last sentence in the second paragraph on page 102 should say ‘bet 24 to 40, that is 3 to 5, that a sixteen year old will die before the age thirty six’.
This is a book for anybody interested in history of mathematics or mathematics in general. You do not need to be a mathematician to appreciate this book.
Overall I would say this is a very successful book and would recommend anybody interested in mathematics or history of mathematics to purchase this book.

Review of ‘The Indian Clerk’ by David Leavitt

I really enjoyed reading this book. It is an excellent account of mathematicians and other academics at Cambridge between 1913 to 1919.
Initially it was difficult to know which parts of the book was fiction and which was fact. It is essentially based on historical facts but with entertaining fiction sprinkled throughout. However it is only when you read the ‘Sources and Acknowledgments’ section you realise the scattering of fiction.
I did like the style of the author and the way he dipped into some mathematics in this book. You do feel that you have understood some of background mathematics behind Ramanujan, Hardy and Littlewood.
However I did not see the need for the visual detail of gay sex in such a book. This should not put the reader off because it is only highlighted in rare instances.
This is a book for anybody interested in history of mathematics or mathematics in general. You do not need to be a mathematician to appreciate this book.
Overall I would say this is a very successful book and would recommend anybody interested in mathematics fiction or fact to purchase this book.

Review of ‘Taming The Infinite’ by Ian Stewart

This is a fantastic book on the history and applications of mathematics. It starts with Hindu Arabic numerals and ends with Chaos Theory. Of course every major mathematical theorem or topic during this period could not be stated in a book of this size and as the author states he had to be selective. My only complaint about the selection is the lack of ‘linear algebra’ because it is perhaps the second most (first being calculus) powerful mathematical tool ever invented.
To fully appreciate this book you must have a reasonable mathematical ability such as a good pass at Further Mathematics A level or equivalent.
Stewart does write in a way that will appeal to most readers and also you can dip into any chapter without digesting the previous chapters. The author has hit the right tone and progression.
A lot of research and time must have been invested into writing this book because of the coverage of applications, the history behind important mathematical developments, profiles of the leading mathematicians etc. I really do like the broad range of mathematical applications throughout the book. The author explains where differential equations are used in the field of physics and modern technology such as radio, tv and commercial jet aircraft and how important Navier Stokes Equation is in fluid mechanics. It goes on to explain where coordinate geometry and trigonometry are used in real life such as graphics, stock market fluctuations, navigation, surveying etc. This is an excellent resource for any A level mathematics teacher who wants to inspire his/her pupils.
The history of mathematics starts with the Hindu Arabic numerals and how they were brought to Europe by Fibonacci. It highlights major historical figures in the mathematics by placing a brief biography in a light shaded grey with an image of the mathematician. However I did not find this sort of feature for Leibniz which is a serious omission since he and Newton founded calculus.
I found the following minor typos:
1. Page 73 the result ‘sin(theta/2)=sqrt(1-cos(theta))/2’ should be ‘sin(theta/2)=sqrt(1-cos(theta)/2)’.
2. On page 156 the statement of Riemann Hypothesis should read ‘complex zeros lie on the line z=1/2 plus or minus it’ not ‘z=1/2 plus it’.
3. Page 260 the statement is written ‘x(t+3)’ should be ‘x(t+ epsilon)’.
This is an excellent book and would recommend that anybody interested in mathematics should purchase this book. The book is a fantastic resource for any college or university library.
Kulδεερ Siηgh
Sunday, 12 April 2009
Homepage at http://mathsforall.co.uk/

Yearning For The Impossible by John Stillwell

Review of Yearning For The Impossible by John Stillwell
This is a wonderful book to read around the subject of mathematics. I learned a lot from this book which can be used in general class discussion to motivate students in a particular field of mathematics. There are some real gems in the book such as:
The word surd comes from the Latin surdus meaning deaf.
The product of sum of two squares is itself the sum of two squares.
A simple example of an elliptic function is the height of a point on an ellipse given in terms of the arc length.
Relationship between Gaussian primes and ordinary primes.
Hamilton’s quaternion which is another number system.
3-D symmetry is rare because only 5 regular polyhedra are fully symmetrical.
If 1 was prime then we would not have unique prime factorisation.
27 is the only cube which is 2 more than a square number.
There are many others like this.
However this book is not for the layman because as the author says in his preface ‘many of the ideas are hard and there is no way to soften them’.
The author has made extensive and fantastic use of illustrations to soften the blow and give an intuitive view of the mathematics. In particular the author has made excellent use of diagrams in the chapter on curved space.
I really do like the historical context the author has placed his mathematics. I did not know that the Chinese had approximated π by the fraction 355/113 which is accurate to 6 decimal places. Nor did I know that the Leibniz series was known to Indian mathematician(s) well before it was discovered by Leibniz. However the author does not give the name(s) of the Indian mathematician(s) who had discovered this series.
The chapter on the fourth dimension has an excellent historical context where the author describes Hamilton’s search for the arithmetic of triples, quadruples etc. The author starts by saying that the great Irish mathematician Hamilton (1805 to 1865) was the first to envisage complex numbers as pairs of real numbers. He goes on to describe the addition of pairs as component-wise addition, that is,
(a, b) + (c, d) = (a+c, b+d) and multiplication as (a,b).(c,d) = (ac-bd, ad+bc) . This system with addition and multiplication of ordered pairs defined in this manner satisfies all the rules of ordinary algebra (field) and the absolute value is multiplicative. The author claims that Hamilton tried for 13 years to extend this idea to triples (a, b, c) but failed. From the ashes of triples he revived a definition for quadruples (a, b, c, d) which ensured that the absolute value was multiplicative and all the rules of ordinary algebra was satisfied apart from multiplication being commutative (skew field). Hamilton called the system of quadruples with addition and multiplication the quaternion.
There is a wonderful chapter on the relationship between primes and Gaussian primes. The Gaussian prime is defined as a Gaussian integer which has an absolute value greater than 1 and cannot be factored into Gaussian integers of smaller absolute value. Numbers such as 1+i and 1-i are Gaussian primes but 2 is not because
2=(1+i)(1-i)
3 is Gaussian prime and an ordinary prime. This means that the ordinary primes are not a subset of the Gaussian primes. The author perhaps should have included that an ordinary prime of the form 1 mod 4 can be written as the sum of two squares but this can be factorised into Gaussian primes because a^2+b^2 = (a+bi)(a-bi). Therefore an ordinary prime of the form 1 mod 4 is not a Gaussian prime.
The last chapter is a nice description of the different kinds of infinity and countable sets. There is a really interesting quote in this chapter ‘ Many people believe the continuum hypothesis put Cantor in a mental hospital’.
In general there are some really good statements in the book which can be used to hook students to study mathematics such as –
‘Calculus as we know it today, is perhaps, the most powerful mathematical tool ever developed’.
‘Infinitesimal calculus brought almost the whole physical world within the scope of mathematics’. However I do have a number of minor quibbles.
Sometimes a proposition or theorem is not clearly stated and it is only when you are half way through a proof you realise what is happening.
There are statements which do not relate to mathematical reality such as ‘square root of a half a turn’ which means a rotation though a right angle. How are the two related?
Are they related by the polar or exponential form of a complex number?
There are a number of places where the book jumps from a simple introduction to the more advanced level within a few pages. An example of this is, we are introduced to the idea of a complex number and within 10 pages we are discussing Bezout’s Theorem.
In places I am not to sure what the author means. For example on page 45 the author says ‘complex analysis is known for its regularity and order.’ Clearly complex numbers don’t have order. For example if i>0 then i^2>0 which is false and if i<0>0 again this is false.
In addition there are a couple of typos such as on page 137 should read c1 not c2 in the formula of line 2. On page 169 it says substitute (7.6) in (7.1) but there is no (7.6).
Homepage at http://www.mathsforall.co.uk/

Review of Basic Linear Algebra by Blyth and Robertson

Review of Basic Linear Algebra by Blyth and Robertson

The book gives a thorough and rigorous treatment of linear algebra which is what a first year student will expect to see on a linear algebra course from a British university.
There are a number of numerical examples which lead nicely to the theory of linear algebra. The authors have hit the right balance between proofs of theorems and techniques to apply such theorems.
The ordering of the chapters is sensible with the first 4 chapters on matrices and linear equations before the more abstract work on vector spaces. The theory and manipulations on eigenvalues and eigenvectors is left towards the end of the book.
A great asset of the book is that it is portable and reasonably cheap at around £16 for students to buy and carry around in lectures and library.
It is also good to see that brief solutions to most problems are at the back of the book.
The only solutions omitted are the assignment problems which the lecturer can set as part of the coursework.
Additionally there are sufficient exercises with good progression and it is good to see a whole chapter devoted to a computer algebra package.
However I have following reservations:
In the introduction to the book it is important to state why linear algebra is critical to the student’s mathematical studies. It should say something like “after calculus the most useful mathematical tool ever developed is linear algebra because it brings the physical world within the scope of mathematics”.
A book on linear algebra should have plenty of illustrations so that the student can envisage what is going on and these illustrations can be used to motivate him or her. This book has a severe lack of diagrams.
More words are required to motivate the student and soften the blow. Each chapter should have an introduction, a list of objectives and a summary. I follow the maxim ‘Tell them, at great length, what you are going to do. Do it, and then tell them what you have done’.
The authors do not write in a way that will appeal to weaker students. It is far too succinct.
The word ‘basic’ in the title is not appropriate for this book. A number of A level students cannot divide 10^(-7) by (1/2x10^4) even with a calculator. I can’t see how students will cope with this book without a serious input by a tutor.
Another issue is that the book is not interactive in any way. It seems to be a one way delivery from the authors to the student. A book like this should include some questions which will make the student think and arouse his/her anxiety. I could not find a single question in the text of the book for the reader. Clearly there are a number of problems for the student to tackle but I am referring to questions such as:
1. Why are matrices important?
2. How can we prove this theorem?

3. What approach are we going to use to solve this problem?
A more serious issue is that once the authors have covered a particular concept they expect the student to fully digest it. This is not my experience of students. I think a particular concept used in chapter 9 which was covered in chapter 2 say, needs to be signposted so that the student knows exactly where the idea was defined earlier in the book.
A less serious issue is that the authors use some very compact and complicated notation. It will difficult for first year students to follow some of this compact notation unless they have seen it before.
The authors use mathematical software, MAPLE 7, but it would have been better to integrate this into each chapter rather than bolt on a chapter at the end. Students will be more confident in using the software if it is used throughout the book.

Homepage at http://mathsforall.co.uk/

Review of Lewis Carroll in Numberland by Robin Wilson

Review of Lewis Carroll in Numberland by Robin Wilson
The book tells the story of Charles Dodgson who is better known as Lewis Carroll the author of various fictions such as Alice in Wonderland and Through the Looking Glass. The book is divided into eight fits and describes how Charles Dodgson was not just the writer of fictions but also a professional mathematician contributing to linear algebra, logic, mathematical puzzles, geometry etc. The book is essentially a biography of Charles Dodgson with a few opening quotes of Carroll’s work.
Charles Dodgson was born in 1832 in Cheshire and studied at Oxford graduating with a first class honours in 1854. One of his hints in studying mathematics was:
“Never leave an unsolved difficulty behind. It is bound to haunt you in some proof or solution later on”.
Wilson also describes in detail the great interest that Charles Dodgson took in photography. He claims that Charles become one of the most important photographers of the 19th Century. The book is sprinkled with some of the images that Charles photographed throughout his life.
It is good to see that the author does not shy away from putting some of the mathematics that interested Charles Dodgson. The mathematics in the book ranges from his defence of Euclid’s Elements to his book on Elementary Treatise on Determinants. However his main interest was in mathematical logic in which he wrote Symbolic Logic which was published in 1896. He also wrote various mathematical puzzles.
Over the last 100 years a lot has been written about Dodgson’s interest in children normally suggesting something disturbing but Wilson refutes all these claims. I do wonder how the political correct will accommodate this refutation with the book containing photographs of young children taken by Dodgson.
Wilson describes how not only is Charles Dodgson a mathematician and an author but also a deeply religious man and a keen walker.
There are some real engaging stories about Charles Dodgson such as when he put a case for a Mathematical Institute at the University of Oxford in 1868. However Oxford had to wait another 65 years before a Mathematical Institute was built.
Another fascinating story the author describes about Charles Dodgson is when his Oxford College (Christ Church) was in financial difficulty. Dodgson proposed that his salary be lowered from £300/year to £200/year. In present day circumstances this would be an unthinkable (or even stupid) act!
In 1881 Dodgson, aged nearly 50, resigned his mathematical lectureship so that he could devote more time to writing books.
Charles Dodgson passed away in 1898 aged nearly 66 in Guildford.
The book is really well written with both characters, Charles Dodgson and Lewis Carroll, being described as a mathematician and an author of fiction.
The book can be hard to follow in places if you are not familiar with A level mathematics but it is possible to skip these parts and maintain the flow of the book. It is a hard balance to strike between putting mathematics into a book like this which can lead to decreased sales and having no mathematics which would be a very serious omission. Robin Wilson has struck the right balance between these two conflicting notions.

Review of ‘The Pythagorean Theorem’ by Eli Maor

This is an excellent book on the history of the Pythagorean Theorem. I learnt a great deal of history of mathematics in relation to Pythagoras’s Theorem. This book is suitable to any student who has basic knowledge of calculus but the layperson will also find it interesting.
The book starts with the assertion that the Babylonians knew Pythagoras’s Theorem 1000 years before Pythagoras but it was the Greeks who proved the result.
There are a number of gems in the book which are not that well known in the mathematics community:
Hypotenuse is derived from the Greek words hypo meaning ‘under’ or ‘down’ and teinen meaning ‘to stretch’. Maor points out the reason for this is that the hypotenuse of a right triangle in Euclid’s Elements was always on the bottom. (I did not know this).
There are over 400 proofs of Pythagoras’s Theorem.
It was the French lawyer ‘Francois Viete’ who first converted verbal algebra into symbolic algebra.
Many more of these gems crop up throughout the book.
Maor does give a number of different proofs of Pythagoras’s Theorem.
More importantly the author does not shy away from producing mathematical expressions and symbols in a popular book like this. Here are a few examples:
1. Every even perfect number is of the form 2^(n-1)*(2^n-1).
2. Viete’s Identity product which expresses 2/π in terms of √2.
3. Shows how the area of one arch of the cycloid is 3 times the area of the circle generating it.
4. Gives an excellent brief description of Hilbert Spaces and non Euclidean geometry.
5. Explains why Pythagoras’s Theorem is not valid in non-Euclidean geometry.
There are many more fantastic mathematical examples. The more serious mathematics is left for the appendices.
Additionally Maor has provided an excellent general history of mathematics such as:
The first woman mathematician was Hypatia (370 to 415).
The University of Gottingen was world renown for mathematics up until the Second World War.
How Edmund Landau (1877 to 1938) shunned all references to geometry. Maor points out that Landau wrote a 372 page book ‘Differential and Integral Calculus’ and it does not contain a single illustration.
How Euler discovered differential geometry but its modern form is due to Riemann and Gauss.
There are also non-mathematical examples of history in the book such as the first European University was Bologna founded in 1088 and why the Christians burned the Library of Alexandria.
You will learn a lot from this book because it has been thoroughly researched and shows the different fields where Pythagoras’s Theorem is used.
The author has also made excellent use of illustrations so the layperson can understand without learning all the details.
Maor has an exceptional method of writing very technical mathematics in a seamlessly way.

Review of ‘The Prince of Mathematics Carl Friedrich Gauss’ by M. Tent

This is a fantastic biography of Carl F Gauss (1777 to 1855). It is a well written book and the author, M Tent, will achieve her goal of inspiring readers to explore the world of mathematics with this book. This is a book for the layman but will be a real inspiration for a sixth former or undergraduate student of mathematics.
The author highlights some well known stories about the young Gauss such as:
By the age of 10 he knew the formula for the difference of two squares and had a smart technique of adding up the first 100 consecutive natural numbers.
The author also claims that Gauss tried to prove the parallel postulate using the first 4 postulates in Euclid’s Elements. Towards the end of the book Margaret Trent shows how this lead to Gauss develop non-Euclidean geometry 25 years before the Russian mathematician Lobachevsky published his work on this topic.
By the age of 11 Gauss could prove the irrationality of the square root of 2.
By the age of 18 Gauss had constructed a regular polygon of 17 sides using unmarked straight edge and compass only.
Some of the material in the book is from Gauss’s diary which has results in shorthand notation such as
∆+∆+∆=N
This result says that any natural number N can be written as the sum of at most 3 triangular numbers. The author claims that this was Gauss’s Eureka moment and was a beautiful discovery.
For his PhD, Gauss proved the Fundamental Theorem of Algebra and also showed the mistakes made by Euler, Lagrange and Alembert in their proofs.
He also made a major contribution to number theory and proved the Fundamental Theorem of Arithmetic.
The author goes on to describe the political turmoil in Germany particularly in the Duchy of Braunschweig (Brunswick). The Duchy of Braunschweig had supported Gauss financially for 15 years but he was killed by Napoleon’s forces in 1806.
After the Duke’s death, Gauss was offered a post at the prestigious University of Gottingen which he took up. He remained there for the rest of his life. At Gottingen he had to teach as well as do research but in the beginning he did not enjoy teaching. He suggested to his wife “I would prefer simply to give my students a text and if they encounter any problems they can see me.” Amongst his students at Gottingen were Richard Dedekind and Mobius. At the University of Gottingen he had become the director of the observatory and published papers on infinite series, astronomy, optics, number theory and algebra.
Additionally between 1833 and 1855 Gauss worked very closely with Weber at Gottingen in the field of magnetism. They also produced the first telegraph system.
Whilst at Gottingen Gauss produced a map of entire kingdom of Hanover by using his triangulation and least squares methods.
Gauss had become an important figure in mathematics throughout Europe and beyond. Even the Universities of Berlin and Petersburg tried to lure Gauss to work for them but he refused each time.
The book is a biography of Gauss with his family and friends at the centre of his life. The only matter of concern is we don’t really know which of the stories are factual.
However this is a smashing book and definitely worth buying.

Review of Impossible? By Julian Havil

This is not a book for the layman. I found this book heavy going and problems with lack of explanation in places.
However there are a number of gems which are definitely worth exploring. These are:
Simpson’s Paradox. This is an example where (a/b) > (c/d) and (p/q) > (r/s) but
(a+p)/(b+q) maybe less than (c+r)/(d+s).
Connection between the continued fraction of 1/e and the optimal number of r out of n.
Why the infinite sum 1/n is called the harmonic series.
Why order is lost in complex numbers.
Moreover there are some fantastic quotes by various mathematicians such as the following by De Morgan:
The ratio of log of -1 to square root of -1 is the same as circumference to diameter of a circle.
This book must be read in conjunction with the author’s other title ‘Nonplussed!’ because he refers to it in a number of places. I have not read ‘Nonplussed!’ but I do think a book like this should be totally independent of any other text.
A major problem with the book is progression is too fast. It is difficult to digest an idea and the author has moved on to higher dimension. A good example of this is the ‘Monty Hall’ problem. The author describes the Monty Hall problem and within a couple of pages he has moved onto various extensions and generalisations of the problem. It would have been better to progress at a slower rate so that the reader understands the initial problem and then is able to follow the extensions on this problem.
In general I found myself taking a lot of time to get through it, because I had to keep going back and looking at things again and again in a bid to understand.
There are a number of typos in particular the brief appendix at the end seems to be full of them. The infinite series for sin and cosine is wrong. It should have alternating signs. There is no fig 4 which relates to subintervals. The proof of log(2) is irrational is incorrect.
This sort of text should have a lot more diagrams.

Review of Letters to a Young Mathematician by Ian Stewart

'Review of Letters to a Young Mathematician' by Ian Stewart is a book which explains why a sixth former should study mathematics at undergraduate. Additionally it is an excellent book to motivate undergraduates to study mathematics at postgraduate level.
However the book contains no or very little mathematics so it is digestible for the general layman. I would have preferred more mathematics in the book because this letter approach could have been a novel way to put over some mathematical concepts.
All the 21 letters start with 'Dear Meg' who is the niece of a factitious mathematician writing the letters. It is set up with a question from Meg (you do not see the question) and the reply from the mathematician. The life span of the letters is about 15 to 20 years starting with the explanation of why Meg should read mathematics at university and ending with benefits of tenure and collaboration. Although the title of the last chapter ‘Is God a Mathematician’ brings in historical quotes such as ‘God is a geometer’ by Plato, ‘God is a mathematician’ by Paul Dirac and ‘God is a Pure Mathematician’ by Arthur Eddington. The book becomes a fantastic collection of letters into the life of a mathematician.
Stewart has quotes sprinkled in his book from the classic ‘A Mathematicians Apology’ by G Hardy. It seems like the book being reviewed is a supplement to the 20th Century Hardy’s classic. Whilst Hardy glorified in his non-applications of mathematics, Stewart shows why mathematics is universal used throughout our lives. He does not make a major distinction between pure and applied mathematics.
Humour is sprinkled throughout the text such as the Dean of a Faculty counting the number of lights in the ceiling of an auditorium. When the mathematician points out that there is no point counting them because there are 8 rows by 12 columns of lights so making it 96 altogether the Dean replies ‘I want the exact number’.
This is an excellent book and definitely worth buying.