European Mathematical Society - prometheus books
https://euro-math-soc.eu/publisher/prometheus-books
enThe Joy of Mathematics
https://euro-math-soc.eu/review/joy-mathematics
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The authors directly address the secondary school student pointing them to mathematical issues that are not covered by traditional curricula. They are of course addressing students in the USA, but most of what they mention applies to the European system as well. I doubt it that most of these young adults will spontaneously read this book for fun, but there are always exceptions of course. Clearly, through these students, the authors are indirectly reaching the teachers, or it may well be the other way around.</p>
<p>
The subtitle of the book: <em>Marvels, Novelties, and Neglected Gems That Are Rarely Taught in Math Class</em> catch the spirit. What are all these tricks, techniques, and theorems which are not usually covered in a regular curriculum because of a lack of time? The authors have organised them in five chapters collecting many of them around a central theme. The first chapter is called <em>Arithmetic Novelties</em>. I hesitate to call these "novelties", unless they are novelties for the student who may read about them here for he first time. The "novelties" are classic arithmetic tools but that may have been forgotten because many computations are performed on computing machines nowadays and not so much in the heads of students anymore. Examples are shortcuts for divisibility checks, formulas to sum numbers or squares of numbers, the Euclidean algorithm, and fun things to know about numbers like palindromic, triangular or square numbers, perfect numbers and the likes, and more material of that style.</p>
<p>
The second chapter collects some algebraic items. Here are some classics like the irrationality of the square root of 2, why a division by zero allows to prove anything true or false, and there are again useful computational methods: the bisection method to find a zero, the Horner scheme for polynomial evaluation, and problems like solving Diophantine equations, generating Pythagorean triples, Descartes's sign rule for zeros of polynomials, and more.</p>
<p>
The geometry topics of chapter 3 take more pages, but that is mainly because these require many graphical illustrations. As you might expect, we find here several less conventional proofs of the Pythagorean theorem and several of its possible generalisations. Also many theorems involve circles (not surprising since the authors published a year earlier in 2016 <a href="/review/circle-mathematical-exploration-beyond-line" target="_blank">The Circle. A Mathematical Exploration Beyond the Line</a>, a book completely devoted to such circle theorems). But there are many other properties as well that involve triangles, spirals, polygons, Platonic solids and star polyhedra, and much more.</p>
<p>
The chapter on probability is relatively short. Here the surprise effect of unexpected results are a central theme. Benford's law, coinciding birthdays, the Monty Hall problem and the related paradox of Bertrand's box, the false positive paradox, and the poker wild-card paradox. Other topics are surprising properties of Pascal's triangle and random walks.</p>
<p>
The last chapter is a collection of miscellaneous problems. About the origin of some of the familiar mathematical symbols, compound interest and the rule of 72 to double your investment, the Goldbach conjecture, countability and the different levels of infinity, properties and constructions of the parabola, the speed of a bicycle as a function of the sprocket wheel used, and several others.</p>
<p>
Anyone who is a bit familiar with the literature on popular and recreational mathematics will find that most items collected in this book are not really novelties, and as a gem, they are not really neglected, but they certainly are rarely taught in math class. However, if you know some teenager who loves mathematics, then this will be a fantastic gift. All the content is up to the level of her mathematics and it are marvels and gems, which are most probably novelties to her. The good thing is that everything is not just raising wonder and surprise, but it is explained why it works and proved when appropriate. It is not a regular textbook tough with formal theorems, proofs and exercises. It is kept at an entertaining level. If, as a teacher, you have some spare time within the strict framework of the curriculum, you can use the book as an inspiration for examples that are stimulating the interest of your pupils.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a collection of popular mathematical topics that are brought at the level of secondary school students but that is usually not included in the regular curriculum because of time constraints. Things are explained and proved at an appropriate level, but it is recreational in the sense that it is not a textbook with formal theorems, proofs and exercises.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/alfred-s-posamentier" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">alfred s. posamentier</a></li><li class="vocabulary-links field-item odd"><a href="/author/robert-geretschl%C3%A4ger" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robert Geretschläger</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781633882973 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 18.00 (pbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">300</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.edelweiss.plus/#sku=1633882977&amp;amp;page=1" title="Link to web page">https://www.edelweiss.plus/#sku=1633882977&page=1</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97a20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A20</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li></ul></span>Mon, 08 Jan 2018 20:58:08 +0000Adhemar Bultheel48155 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/joy-mathematics#commentsBlockbuster Science: The Real Science in Science Fiction
https://euro-math-soc.eu/review/blockbuster-science-real-science-science-fiction
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Science and science fiction (SF) are not too far apart and the boundary can become fuzzy in some cases. Terms and ideas now generally accepted were used for the first time in fiction novels. The word "robot" is an example of such an SF product and Jules Verne's space gun is a well known prediction of Apollo 11 used to realize the moon landing realized much later. Where science reaches its current boundaries, fiction can extrapolate and take these results to the next level. Exploring these boundaries is the purpose of this book. There is virtually no limit to the fantasy of fiction writers, but somehow the adventures of their heroes and villains take place in worlds that are inspired by our familiar society and by the environment and technology that we know today. Hence the science in SF involves almost anything we know or struggle with today: from quantum mechanics to biology, to artificial intelligence, to cosmology, to relativity theory, and of course, some ethical and philosophical problems can arise when science is pushed a bit further. All this science can be discussed on a very technical level and that can involves deep mathematics and complicated physics. So it is a challenge to discuss advanced science topics and yet avoiding all the difficult technical details.</p>
<p>
Bernstein has written some science fiction himself and he is professionally involved in economics, statistics, and mathematics as a managing consultant. So he is well earthed to real science and has the fantasy to fictionalise it beyond its boundaries. Testimony of his scientific knowledge is the astonishing amount of well documented technical knowledge that he summarizes in this book. Most of this science is based on mathematics, but as he writes in the introduction:</p>
<blockquote><p>
Also, don't worry about the math that occurs here and there, because these references are very limited. Never fear math. It is the language of science. In fact, as with spoken languages, it is fraught with tongue twisters the scientists sometimes take too seriously.</p></blockquote>
<p>
Faithful to his promise, there is indeed not much mathematics explicitly present, and neither does he become technical about the physics. Nevertheless he starts with a discussion of the two pillars of twentieth century physics: quantum mechanics and relativity theory. These two chapters are characteristic in approach and format for the other chapters (there are 21 chapters, which is just half of 42; it might be a coincidence but 42 is, according to Douglas Adams, the answer to the ultimate question of life). Most of the chapters are short chunks, hashing up the complex themes in digestible bites. They place the real science results against the fictional extrapolations. For example spacetime has black holes which may be used for time travelling, or they may create wormholes which would allow just travelling from on point in spacetime to another without violating the speed limit of light. While these are theoretical constructs in real science, they are mostly presupposed trivialities in science fiction. Both chapters have so-called bonus material for example discussing the twin or the grandfather paradox of time travelling or using Einstein's formula to compute the amount of energy that is packed in a human body, or to explain what quantum suicide and quantum immortality means in a multiverse context. Other chapters have "parting comments" which are just take-along summaries of what has been discussed in that chapter. The book has also three "interludes" which are chapters that deviate from the standard format. These discuss only science, there is no fiction. They briefly introduce some basics: the first one is about atomic theory, the second about transhumanism, that is when the human body and brain are biologically and technically upgraded until the result can hardly be called human anymore, and the third is about mass, dark matter and dark energy.</p>
<p>
</p>
<p>
The broad spectrum of themes discussed include string theory and extra dimensions, the vastness of our universe, parallel worlds, energy resources, the origin of life, genetic modification and cyborgs, global warming and other catastrophes, colonisation of the galaxy, computers, robots and AI, extraterrestrial life, materials engineering, virtual reality, and possible ends of the universe as we know it. All of these have shown up in science fiction media (comics, stories, books, films) in some form and Bernstein gives several concrete examples. Everything he claims is well documented with references (to both the scientific and the SF literature). These references are collected chapter by chapter at the end of the book. There you can also find a useful glossary explaining many technical terms from "abiogenesis" and "absolute zero" to "wobble method" and "zombies" but also the useful index and extensive lists of SF literature, movies and songs. Extensive as the latter lists may be, it is obviously reflecting a selective choice made by the author, because the amount SF literature, movies, and TV-series is too vast to aspire any degree of completeness.</p>
<p>
So there is no explicit mathematics and there are no formula in the book, but many mathematicians work in applied areas of AI, computer science, theoretical physics, materials science, nanotechnology, and many other engineering applications. So I am sure there are enough geeks among them that love science fiction and thus will probably love this book. The author did an amazing lot of fact checking and he has ample illustrations from SF. His style is really crisp, up tempo, and to the point, but most of all I love the humour he uses. An example: where he explains the spaghettification effect when entering a black hole he writes:</p>
<blockquote><p>
Think of an event horizon as a fence with a big Keep Away sign hammered into it. Personally, I would do what it says. [...] If you do decide to ignore the warning and trespass on the event horizon, I hope you like pasta. This is not an adventure I would recommend. However, if you insist, the first thing you do is put on the latest spacesuit ad disembark from your starship. There is no reason to endanger the rest of the crew.</p></blockquote>
<p>
But don't be mistaken, the technical or philosophical material is serious, and he is not joking much when he discusses global warming, a hot topic these days, that unfortunately is no fiction. Even the geeks among the readers may learn a thing or two that is new to them, and it is also an interesting way to detect some new SF literature or movies you didn't know about. For potential authors who want to start writing their SF novel and they want it to be hard SF, they better check out all the facts that are provided here. If you can't get enough of this, I can refer to a similar book by Charles L. Adler <a href="/review/wizards-aliens-and-starships-physics-and-math-fantasy-and-science-fiction" target="_blank">Wizards, Aliens, and Starships</a> who also includes the fantasy literature. Also Paul Nahin wrote about the topic in <a href="/review/holy-sci-fi-where-science-fiction-and-religion-intersect" target="_blank">Holy Sci-Fi!</a> but he is more focussing on the religious aspects and less on the science.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Bernstein explores in this book the boundaries between what is scientifically known today and what science may be capable of in the future and what will always be science fiction. Using many references from the scientific literature and the SF media (books and film) he succeeds in separating the science from the fiction. He has a scientific background and has written some SF himself, but he succeeds in discussing scientific topics avoiding the mathematical and technical material.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-siegel-bernstein" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Siegel Bernstein</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781633883697 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 24.00 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">304</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.penguinrandomhouse.com/books/557922/blockbuster-science-by-david-siegel-bernstein/9781633883697/" title="Link to web page">https://www.penguinrandomhouse.com/books/557922/blockbuster-science-by-david-siegel-bernstein/9781633883697/</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a69" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a69</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a79" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a79</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/70-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/81-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-01</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/83-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">83-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/85-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">85-01</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/92-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">92-01</a></li></ul></span>Mon, 08 Jan 2018 20:29:58 +0000Adhemar Bultheel48154 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/blockbuster-science-real-science-science-fiction#commentsThe Forgotten Genius of Oliver Heaviside: A Maverick of Electrical Science
https://euro-math-soc.eu/review/forgotten-genius-oliver-heaviside-maverick-electrical-science
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Oliver Heaviside (1850-1925) was a British self-made electrical engineer and mathematician. He is probably best known as an electrical engineer, although his name is not explicitly attached to many terms. There is a Heaviside condition for transmission lines, but maybe less known is that he also coined terms such as conductance, inductance, impedance, and many more. In mathematics, his name is explicitly attached to the Heaviside step function usually denoted as $H(x)$ in his honour, although he himself preferred the notation $\mathbf{1}$ instead. Sometimes his name is also attached to a method to compute the partial fraction expansion of a rational function. However, probably his most important contribution to science is that he reformulated the Maxwell equations in the form as we know them today. As a side product he introduced vector calculus and the associated (force)fields. It also brought complex numbers and complex analysis into electro-technical formulas. He is also the originator of operational calculus. The mathematics community was originally reluctant to accept it because it lacked fundamental rigidity. But it worked so well that it could not be ignored and others provided the necessary rigidity. It allowed to transform a differential equation into an algebraic one, which is much easier to solve. The letter <em>p</em> that is often used as the variable in the Laplace domain was his notation. In the time domain, it is a differential operator. The square root of $-1$, which mathematicians usually denote as $i$ is denoted as $j$ in (electrical) engineering because $i$ or $I$ was used for current (although Heaviside preferred to use $C$ for current). He considered $j$ as an operator that had the effect of delaying the signal with a quarter of a cycle, just as mathematicians see a multiplication with $i$ as a rotation over 90 degrees. These are but a few illustrations to show that his field was electrical engineering, and that he is more recognized for his legacy in that domain, it can be said that his influence on mathematics, although less known, is equally important.</p>
<p>
Mahon has used the term "maverick" in the subtitle which is most appropriate in the case of Heaviside. Now one can be a maverick in different ways. For example Richard Feynman was really unconventional and did not pander to the customs attached to his status, but it was all in a playful and friendly way. Heaviside on the other hand was a grumpy, stubborn, loner, who did not shy away from aggressive reactions and including insulting remarks in his scientific papers about people he did not agree with. He was for sure not the most likeable person. He had only few friends and admirers who tried to mediate between him and the scientific community. Among them were George FitzGerald, Oliver Lodge, and Heinrich Hertz, who together with Heaviside became known as <em>The Maxwellians</em> by the book of Bruce Hunt (1991). No wonder that such an outspoken character has inspired other biographers to write about Heaviside. Fortunately much of his publications, notes and letters are at their disposal to reconstruct his personality. Paul J Nahin's <em>Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age</em> (John Hopkins University Press, 2002), is a basic reference, and there is an account of Heaviside's character by his friend G.F.C. Searle described in the booklet <em>Oliver Heaviside, the Man</em> (1987). Also the book under review was published earlier in 2009 by a different publisher and with a slightly different title <em>Oliver Heaviside: Maverick mastermind of electricity</em>. Not much is changed in this edition except the spelling and the notes at the end which have been extended.</p>
<p>
As a child Oliver Heaviside had scarlet fever which left him partially deaf. This may in part explain his tendency to withdraw from crowds and prefer solitude. He suffered of several illnesses throughout his life, some were due to poverty and negligence. His uncle was Charles Wheatstone (from the Wheatstone Bridge known in electricity) and an expert in telegraphy. To understand Heaviside's scientific breakthrough, one should think back mid 19th century. James Maxwell just discovered the relation between electricity and magnetism which he presented as a complex system of 24 equations. Submarine cables for telegraphy were laid with a lot of experimentation and disastrous failures. Wheatstone looked after Heaviside's education, but when his parents could not afford the studies anymore he studied on his own. When working for Wheatstone's telegraph company he trained himself as an electrician and published a paper about the Wheatstone Bridge, that was received well by, among others, William Thomson (Lord Kelvin) and James Maxwell. His next publication earned him his first enemy: R.S. Culley, the engineer-in-chief of the nationalized telegraph company. Heaviside's view on the duplex method was opposed to Culley's and he ridiculed the man for his short-sightedness. His next achievement was a development of a theory for transmission lines (mathematically these are the telegraphers equations) which are of course tremendously important for telegraphy (and telephony) cables. The speed of light popped up in his equations, which pointed already to the electromagnetic interpretation of light.</p>
<p>
Heaviside has been poor throughout his whole life. He got a small income form his scientific contributions to <em>The Electrician</em> in the period 1882-1902. They also published his 3 volume work <em>Electromagnetic Theory</em> (1893-1912) but that didn't earn him much money. He had moved to London in 1882. It was there that he reduced twelve of Maxwell's equations to just four using vector calculus. They are in their simplest form the following relations: $\mathrm{curl}\,\mathbf{E}=−\frac{\partial \mathbf{B}}{\partial t}$, $\mathrm{curl}\,\mathbf{B}=\frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t}$, $\mathrm{div}\,\mathbf{E}=0$, and $\mathrm{div}\,\mathbf{B}=0$, where $\mathbf{B}$ is the magnetic field and $\mathbf{E}$ the electric field, and $c$ the speed of light. A mathematical beauty by its simplicity and its symmetry. Previously people had tried to deal with them using quaternions, invented by Hamilton in 1843. Together with his brother Arthur, Oliver had worked on the design of a distortion free transmission line. One had to arrange that $G/C=L/R$, which is known as the Heaviside condition ($C=$ capacitance, $R=$ resistance, $L=$ inductance, and $G=$ shunt inductance). This was his gift to society that would make long-distance telephony possible. However Arthur and Oliver had to ask for publishing permission from their employer the Post Office, but that was surprisingly refused. The bad omen was William Preece, who had opposing views on the solution and happened to be the engineer-in-chief of the Post Office then, and this resulted in a lifelong and bitter battle between him and Oliver. Oliver referred to Preece's ideas as the "drain-pipe theory". Oliver's results were eventually published with some delay, which brought him new fame. Unfortunately for Oliver, Preece was well respected and the next year he became president of the IEE and in this position he could thwart Oliver some more. In that period the Maxwellians became friends and collaborators. First FitzGerald and Lodge, and later Hertz from Germany.</p>
<p>
Approaching the turn of the century, fate still haunted Oliver. His mother died in 1894, his father in 1896. Poorer than ever and plagued by illness his friends organized a pension for him and he moved to live on his own. Pupin (and AT&T) in the U.S. got rich on a patent for distortion-free transmission lines based on a formula that Heaviside published 3 years earlier. When the Royal Society wanted to award him the Hughes Medal, he refused because they had rejected the last part of his paper on operators in physics saying that the mathematics were not rigorous enough. Nevertheless, later he accepted a pension, just to survive. But recognition started to emerge. At some point he was even shortlisted for the Nobel Prize (like Einstein and Planck, but the 1912 Prize went to Niels Dalen). By then he had published his three volumes of his <em>Electromagnetic Theory</em>. He got an honorary membership of the American IEE (1918) and he was awarded the Faraday Medal of the IEE (1921). When he was found unconscious in 1925 he was moved to a home, but he died shortly after.</p>
<p>
Mahon goes through the life of Heaviside in twelve chapters, each one corresponding to a place where Heaviside lived and covering successive time periods. However each chapter is also the starting point to discuss one of his achievements. Because that requires to explain what preceded, how he came to his conclusion, and how and by whom the idea was picked up, and what its eventual fate was, the period covered in a chapter is much broader than announced in the title. Hence, the text is not always strictly chronological. Therefore the chronology summarised in the time-line in the beginning of the book comes in handy. There are also, sometimes relatively extensive, biographies of the persons who played a role in Heaviside's life, and then the list of the main characters inserted after the time-line is also useful. It is clear that Mahon, who also authored a book on Maxwell and co-authored another one on Maxwell and Faraday, is an admirer of Heaviside's electrical contributions, but he also gives credit to his significance for mathematics. The book is obviously written for a general public, so the discussions about Heaviside's results are only slightly technical. There are many quotations from texts written by Heaviside, which explains a lot of how his character was, and what he thought and expected from others. The way he wrote about his housekeeper at a later age is hilarious. He must have been a very difficult man to live with. There are also 33 pages with extra notes giving additional explanations. The book gives some insight in his work and the man that is behind it. Heaviside's very peculiar and enigmatic character will forever remain inscrutable, but Mahon does a really good attempt to understand what was driving this lone genius and to give him some of the respect that he rightfully deserves and that he had to miss during most of his life.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is the second edition (only slightly modified) of a biography of Oliver Heaviside (1850-1925). Heaviside was a self-taught electrical engineer and mathematician. He had an obtrusive character and an unconventional approach of doing research. Therefore it took a while before the geniality of his work was recognized.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/basil-mahon" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Basil Mahon</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2017</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781633883314 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 29.00 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">336</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematical-physics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematical Physics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.penguinrandomhouse.com/books/556320/the-forgotten-genius-of-oliver-heaviside-by-basil-mahon/9781633883314/" title="Link to web page">https://www.penguinrandomhouse.com/books/556320/the-forgotten-genius-of-oliver-heaviside-by-basil-mahon/9781633883314/</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/35q61" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">35Q61</a></li></ul></span>Mon, 08 Jan 2018 19:59:11 +0000Adhemar Bultheel48153 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/forgotten-genius-oliver-heaviside-maverick-electrical-science#commentsThe Circle. A Mathematical Exploration Beyond the Line
https://euro-math-soc.eu/review/circle-mathematical-exploration-beyond-line
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Alfred Posamentier has (co)authored a few dozen books on mathematics and mathematics education. Several of these are intended to popularize math. In 2012 he co-authored with Ingmar Lehmann a book on triangles <em>The Secrets of Triangles: A Mathematical Journey</em>. Triangles are about the simplest mathematical objects interesting enough to prove many theorems about, and they have many applications. Think of the fact that any polygonal area can be triangulated and if the triangular net is made fine enough, they approximate and subdivide any area or surface.</p>
<p>
In this book, the triangles are replaced by circles as the key objects. It contains, among other things about circles, a large variety of classical and less known —but nevertheless quite interesting— theorems that can be formulated concerning properties of circles. These involve often cyclic polygons (i.e., whose vertices lie on a circle) or polygons whose sides are tangent to it. It will not come as a surprise that also here triangles still play a prominent part in this game too.</p>
<p>
In a first chapter some elementary properties and definitions are recalled. It may be a surprise for some readers that the circle is not the only 2D object that has constant breadth. This property it has to share with the Reuleaux triangle, an "inflated" equilateral triangle that is the intersection of three circles with centers at the vertices and radius equal to the side length. A slightly flattened version of it found an ingenious application in the design of the Wankel engine.</p>
<p>
Chapters 2 and 3 discuss a first set of theorems concerning circles. In most cases this starts with a known theorem like for example Ptolemy's (in a cyclic quadrilateral the product of the diagonals equals the sum of the products of the opposite sides). In the case of a rectangle this reduces to Pythagoras' theorem. Generalizations consider n-gons whose vertices are on a circle. This kind of strategy to discuss a theorem is repeated for other theorems: a formulation of the theorem is given, mentioning its origin and sometimes a few sentences about the mathematician that is behind it, then a proof, and by considering special cases, or sometimes generalizations, seemingly unrelated theorems turn out to be included as well. Almost always, there is a further exploration of the problem involved. For example if the theorem claims the collinearity of 3 points, then it may be followed by an analysis of properties about circumferences or areas of triangles and/or circles that were involved in the proof. In this way we are guided along the theorems of Simson, Miquel, Pascal, Brianchon, Ceva, the butterfly theorem, the nine point circle theorem, the six and the seven circles theorems, the Gergonne point theorem, Poncelet's porism, the arbelos, and Ford circles. When a proof is too complicated, it is not included (notes and references are at the end of the book), and sometimes more technical stuff is moved to an appendix, but most proofs are rather easy to follow and are brought in a reader friendly way with many figures that illustrate the successive steps to be followed in the proofs. So proofs are not hard abstract manipulations of formulas, but they rely heavily on the visualization. The text merely explains what the successive steps are, where for example the equal angles are, or the similar or congruent triangles are and why this should be true. It still requires some mental flexibility and an elementary geometric knowledge to follow each step of the proof but it is easy going.</p>
<p>
Circle packing (in a confined space like a rectangle, a circle, or a triangle) is discussed in a short chapter 4. It is explained how it is related to an application in a computer program <em>TreeMaker</em> to design origami patterns.</p>
<p>
The next 4 chapters deal with geometric constructions. Equicircles (the 4 circles that are tangent to all 3 sides of a triangle, 1 inside and 3 outside the triangle) get their own chapter with a computation of their centers and radii.<br />
Chapter 6 is a discussion of the Apollonius problem. That is how to construct a circle that is defined in different ways by points, lines and other circles like containing 3 given points (PPP), or two points and a tangent line (PPL), or a point and 2 tangent lines (PLL), 2 points and a tangent circle (PPC), etc. There are 10 such possibilities. All constructions should be done with straightedge and compass.<br />
The next chapter introduces reflection in a circle. This transforms circles and lines (i.e. circles with infinite radius) into circles and lines. Some of the previous Apollonius problems can be solved in the reflected setting. It can also help in the construction of Steiner and Pappus chains.<br />
Finally chapter 8 is discussing Mascheroni constructions. The ancient Greek tradition allowed to use only straightedge and compass to do all the geometric constructions. It is impossible to do everything with only the straightedge, but one can do without it as was proved by Mohr and independently by Mascheroni. Of course we cannot draw a straight line with a compass, but one can construct any point on a specified line whenever it is needed. Of course the constructions are more involved, but it is shown that with 5 fundamental constructions everything can be done without the straightedge.</p>
<p>
Chapter 9 is again a short interruption from the mathematics since it gives a very brief survey of how the circle was used in arts, in shaping the landscape, and in architecture. This topic could easily be the subject of a whole mathematical picture book on its own, but I do not think the authors have the intention of being complete here. There are just a few examples and it serves to relax a bit from the mathematics in the previous chapters and in the two chapters to follow.</p>
<p>
The remaining two chapters are indeed back into mathematics, but mainly descriptive.<br />
Chapter 10 is for example more recreational: no proofs, but still many graphs. It's all about circles rolling along a line or a circle: the cycloids, hypocycloids, epicycloids, and related curves. Several of these come with a history like the Aristotle wheel paradox and of course the invention of the wheel itself. For the playful aspect, the Spirograph is clearly the instrument of choice. This chapter is not by the authors but it is contributed by Christian Spreitzer.<br />
The last chapter is about spherical geometry. The circle is the only curve that fits also on a sphere. It is well known that the shortest path between two points on a sphere follows a great circle through these points. This explains the route followed during transatlantic flights. There is also the spherical triangle whose angles can sum up to just below 540° and the counterintuitive hight to which a rope around the equator will rise when its length is increased by 1 meter.<br />
An afterword is contributed by Erwin Rauscher who gives a cultural introduction to the circle. The author is different, but it would easily blend in with Chapter 9 on the use of the circle in arts.</p>
<p>
Like several of Posamentier's previous books, this is a book mostly about mathematics, but the "gentle" version, painted on a cultural and historic canvas. The proofs stress the importance of the visual aspect. I am afraid that much of this geometric kind of reasoning has nowadays been largely replaced by algebraic manipulations which, in my opinion, is regrettable. I am old enough to have had this geometric education in secondary school still largely influenced by the Greek tradition of Euclid's Elements, and I remember how I enjoyed solving these geometric "puzzles". It may be one of the reasons that made me decide to become a mathematician. I truly enjoyed re-living this happy experience of my youth by reading this book since during my math studies at the university and in my later career I never used or needed this kind of geometric argumentation.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Like in several of his previous books, Alfred Posamentier, this time in collaboration with Robert Geretschläger, brings this "gentle" kind of mathematics for a broader public. This time it are reflections on geometric problems that are sketched on the canvas of a cultural and historic background. There are proofs but these rely strongly on the many graphics and the geometric constructions. Therefore the mathematics stay away from the dull abstract algebraic formula manipulations that may repulse students.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/alfred-posamentier" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Alfred Posamentier</a></li><li class="vocabulary-links field-item odd"><a href="/author/robert-geretschl%C3%A4ger" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robert Geretschläger</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">9781633881679 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 25.00 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">349</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.penguinrandomhouse.com/books/539658" title="Link to web page">http://www.penguinrandomhouse.com/books/539658</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/97-mathematics-education" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97 Mathematics education</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/97g40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97G40</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/51-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/52c26" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52C26</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/00a66" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a66</a></li></ul></span>Fri, 02 Dec 2016 12:34:45 +0000Adhemar Bultheel47310 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/circle-mathematical-exploration-beyond-line#commentsEinstein at Home
https://euro-math-soc.eu/review/einstein-home
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Friedrich Herneck (1909-1993) is a German science historian from the DDR period who wrote among other books also a biography of Albert Einstein. In 1978 he published a book <em>Einstein Privat</em> which contained the transcript of five interview sessions he had with Herta Schiefelbein, who was the household helper of the Einstein family in the period 1927-1933. It had also a short introduction about Herneck written by Dieter Herrmann. This book brings the English translation of the German original. It is translated by Josef Eisinger who also added to his translation an Einstein biography of 36 pages to sketch the necessary background.</p>
<p>
In the period covered by the interviews, Einstein was already famous all over the globe. In 1927 he was officially divorced from his first wife Mileva Marić in 1919 who then continued to live in Zürich with their two sons (Hans Albert and Eduard). She got Einstein's Nobel Prize money and half of his salary as alimony. Einstein married his niece Elsa Einstein right after the divorce. He and Elsa had an apartment in the Haberlandstrasse in Berlin. She had 2 daughters from a previous marriage. The eldest married already in 1924 but the youngest Margot lived with them until she married in 1930. In 1928 Einstein was overworked and was sick with heart problems for almost a year. In 1929, when Einstein turned 50, the Einstein family moved for a short while to a house in Caputh (near Potsdam) close to the place where their new summer house was built. The construction of this house was one reason, but it was also partially to flee the birthday ceremonies and the "personality cult" as he called it. They moved to the new building with view on the nearby lake later that year. Till 1933 they lived there part of the year and spent the rest in Berlin or they were traveling (twice to Pasadena where Einstein lectured in CalTech). In 1933 Einstein left Germany definitively.</p>
<p>
Eisinger's biography covers the whole life of Einstein, but the previous paragraph sketches a brief summary of the part that situates the period in which Herta was insider to the Einstein household.</p>
<p>
The interviews themselves are transcribed as questions (by F.H.) and answers (by H.S.). In his questions, and as a reaction to Herta's answer or when she does not remember, Herneck often adds additional comments from other biographers or other sources. For these, references and further details are often given as notes in an appendix. In the later interviews they are sometimes inserted as intermissions in the interview. The conversation is rather formal and somewhat old-fashioned but all possible details are covered. A detailed description of the Haberlandstrasse apartment: the ground plan, the furniture, the frames on the wall, everything, (repeated for the summerhouse in Caputh), what was included in the tasks of Herta, how she had to dress, what utensils she had at her disposal, how she was addressed by whom, etc. Obviously also many questions about the members of the family, what they used to eat and when, which visitors came and how often,...</p>
<p>
So we learn that in that period Einstein carried very little money with him, he was not interested and very negligent in his clothing, he drank only caffeine-free coffee, he did not adhere to a strict daily schedule, he smoked the pipe a lot, he ate many vegetables with little spices and liked especially mushrooms and had several eggs for breakfast, he enjoyed playing the violin, sometimes while thinking about a problem, his English was not very good, but he had keen interest in Esperanto,...</p>
<p>
In Caputh, Einstein owned a boat that he had received as a birthday present. He loved sailing very much and ... he had a weakness for beautiful women and flirted with several. The blonde Austrian Margarete Lebach was one of them. When he went sailing on his boat with Grete, Herta recalls some "loud discussion" between Einstein and Elsa. Later on Elsa settled in the situation and tried to avoid the troublesome situation and she left for Berlin every time she knew Grete was coming.</p>
<p>
The last interview is concerned with the period when the Nazis took power in 1933. Einstein resigned from the Prussian Academy of Sciences and accepted a job at the Institute of Advanced Studies in Princeton. Here it is more Herneck who is telling the story of what was going on since Herta was not aware of most of these events at that time. However Herta is interviewed about what happened in the Berlin apartment and with Elsa's children. Margot and her husband had left for Paris but Ilse and her husband stayed in Berlin and took care of Einstein's material. They employed Herta for a number of weeks before they too left Germany. The <em>Kriminalpolizei</em> came looking after Margot's husband in the city apartment but fortunately they had left already. Somewhat later a group of men in uniform came back for a house search when Herta and Einstein's secretary were packing up things to leave the apartment. The place was robbed of the pictures and the carpets. Remaining things and furniture was shipped to Princeton. Herta was interrogated by the <em>Kriminalpolizei</em> about "the Jew Einstein with an Aryan servant" but she had nothing to say in his disadvantage. Nevertheless all Einstein's possessions were confiscated anyway.</p>
<p>
Not all of the information exposed in the interviews is really exciting, but at least it gives a minute description of how the Einsteins lived in those six years. This account stays far away from science and mathematics and tells about a day-to-day family household. One can hardly say that it was the household of ordinary people, given Einstein's celebrity, but still it was a family of people who each had their own character and idiosyncrasies. Herta certainly gives a happy and flattering picture and has her own description of all the celebrities that came visiting. Herneck's comments are a bit preachy but I take it that that was the common interviewing style in those days. </p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Herta Schiefelbein was a maid in Einstein's household from 1927 till 1933. Friedrich Herneck was a historian who wrote an Einstein biography and interviewed Herta Schiefelbein about her job in the Einstein family. These interviews were published in German as a book entitled <em>Einstein Privat</em> in 1978. This book contains the translation by Josef Eisinger who also added a short Einstein biography to provide the necessary background</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/friedrich-herneck" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Friedrich Herneck</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1633881464 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">11.93 USD</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">200</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.amazon.com/Einstein-at-Home-Friedrich-Herneck/dp/1633881466" title="Link to web page">https://www.amazon.com/Einstein-at-Home-Friedrich-Herneck/dp/1633881466</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01a75" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A75</a></li></ul></span>Mon, 08 Aug 2016 12:04:19 +0000Adhemar Bultheel47099 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/einstein-home#commentsA Numerate Life: A Mathematician Explores the Vagaries of Life, His Own and Probably Yours
https://euro-math-soc.eu/review/numerate-life-mathematician-explores-vagaries-life-his-own-and-probably-yours
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
John Allen Paulos is mathematics professor at Temple University, a columnist, media figure, often invited for public lectures, an intensive twitter, and author of several popular books on mathematics. His first book <em>Innumeracy</em> (1988) in particular is well known in which he exposes the dangers of mathematical illiteracy and misconceptions among common people. Other books deal with mathematics in relation with stock markets, humor, and religion.</p>
<p>
From reading the introduction, it was already clear to me that I was in for something different, something besides the ordinary of a popular math book slash autobiography. And indeed after I finished reading, it is difficult to classify or characterize the contents. Perhaps it could be defined as a book about an attempt by a mathematician to write a partial autobiography, and perhaps somewhere the word numerate should also be smuggled in too. There are some chunks of an autobiography indeed, and they are in a rather chronological order. And then there are also almost philosophical reflections upon what it is like to write a biography or an autobiography. Is it even possible to tell a life story? How reliable can a biography be? In fact, as he did in his previous books: apply mathematics to unmathematical questions, he applies here mathematics to questions that can (and should) be asked when writing a biography. There are excursions about mathematics too of course, often drawing parallels between mathematics and some other event or situation or concept. At the end of the introduction he gives his cv in about 10 lines. These are the facts, and although we learn something about the personality behind these facts, there are not many essential facts added while reading the rest of the book.</p>
<p>
While telling some anecdotes from his childhood, he applies his mathematics like in <em>Innumeracy</em> to several of the items like for example the probability of obtaining a missing baseball card. Or think about this problem: Cut 1/2 inch from all sides of a 10 inch cube and the remainder is? Only about 73%! Most people would spontaneously estimate a much larger part. This trick of smuggling in the numerics is repeatedly applied in other chapters too. He has a particular fascination for the number e. Problems whose result is related to e keep showing up. Paulos also displays his sense of humor. Sometimes the joke is mathematical or logical, style Groucho Marx, who is one of his favorites. When Paulos was a guest in a tv-show, he was interviewed by a model (more beautiful than numerate), she was reading the question from a poster behind him, he waited till the text disappeared and then asked her to repeat the question. Mumbling and stumbling she tried to reformulate it. Afterwards the producer thanked him for exposing something to the manager that he himself had suggested several times before. .</p>
<p>
There is some mathematics is every chapter. Several chapters deal largely with statistics. For example, it is observed in practice that the normal distribution does not hold near the pass/don't pass grades but shows some discontinuous drop. The average person does not exist, in fact everybody is abnormal. More statistics in a later chapter on dating (a well known rule says that you have to take the next best after rejecting the first 37% of the candidates and he notes that 37% is approximately 1/e). And when after dating you decide on living together you might wonder what is most effective, leaving the toilet seat up or down?. More statistics are needed to analyze the probability of (seemingly unlikely) coincidences.</p>
<p>
And there is more math stuff. There is the problem of self-reference in an autobiography. Writing the autobiography affects the author, leading to problems of infinity and the continuum hypothesis. Life is also very complex, depending on a zillion of parameters, so that a biography is automatically an approximation problem. The butterfly effect may result in a completely different outcome of anybody's life. Here he attaches the anecdote that he wrote in one of his columns that the margin of error was larger than the margin of election referring to the balance of the votes in Florida during the presidential election in 2000. A judge cited him and ordered re-counting the votes, resulting in the election of George W. Bush, which he still regrets.</p>
<p>
How come there seems to be some tendency to have more memories from your childhood than from the central part of your life. Is that a kind of Benford's law? But there are some turning points in his middle life and these he surely remembers: marriage, becoming a father, Bertrand Russell who answers to one of his letters, the publication of his first book etc.</p>
<p>
For the biographer, there is also the problem of quantitative information. With a quantified selver like Stephen Wolfram, numbers is no problem at all. Paulos doesn't even have a Facebook account, but he belongs to the Twitter community, an occasion to do some mathematics again: he compute the number of possible tweets, and he explain some network analysis defining hubs and authorities and explains the small world phenomenon as a result of network distances like Erdős and Bacon numbers.</p>
<p>
Paulos had a bad experience with stock markets because he lost a lot of money through the WorldCom debacle. He wrote a book <em>A Mathematician Plays the Stock Market</em> about it and also here this episode is represented. Just like also his other books are shimmering through in this one at several places.</p>
<p>
Perhaps lives are too complex, too fractal, too multilayered, and a person cannot be completely understood (a digression on Chaitin's algorithmic information theory), hence can not be caught in a biography. Life goes on in his children and grandsons and after thinking about mortality (Gompretz' law), he reflects on topology (lives can be very different and yet still be conformal) and Brouwer's fixed point theorem (completely different lives may still have a common fixed point where they meet). As life tends to its end, it approaches a steady state, meaning that highlights become less frequent. After some reminiscences about his father, he concludes with the empty set, a set from which all the integers, and rationals and reals and in fact all mathematics can be created.</p>
<p>
I really did enjoy reading this book. Paulos writes very eruditely, although his thoughts sometimes wander away from the main line, there are wonderful things said in many of his excursions. He is philosophical, charming, and funny in a gentle way. I don't know if all the mathematical puns will be understood by non-mathematicians, but for mathematicians, it is a wonderful and playful (self-)reflection on mathematicians, their mathematics,... and their biographers. </p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
In his previous books, Paulos has applied mathematics to everyday facts and problems, to financial matkets, to religion, etc. In this book he reflects on what it means to write a(n) (auto)biography and he checks his numeracy and applies mathematics to the task, or he rather shows some mathematical parallels with the task of writing an autobiography, an authography of a numerate mathematician who has warned his readers about their dreadful lack of numeracy. He has tried his whole life to do the opposite and as we learn in this book, so far it has been a really numerate life indeed. <br />
</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/john-allen-paulos" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">John Allen Paulos</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1633881181 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">14.83 USD</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">200</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.amazon.com/Numerate-Life-Mathematician-Explores-Vagaries/dp/1633881180" title="Link to web page">https://www.amazon.com/Numerate-Life-Mathematician-Explores-Vagaries/dp/1633881180</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a05</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</a></li></ul></span>Sat, 30 Jul 2016 06:26:10 +0000Adhemar Bultheel47086 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/numerate-life-mathematician-explores-vagaries-life-his-own-and-probably-yours#commentsThe call of the primes
https://euro-math-soc.eu/review/call-primes
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is another book written to promote and popularize mathematics to laymen and unbelievers. In most cases, the author of such books is a mathematician who has to be careful not to introduce technical terms without explanation. This time the author is not a mathematician, which helps of course to guarantee that the book really stays at a level accessible to anyone. As an example, at some point it is even explained what a difference is between a conjecture and a theorem. Owen O'Shea is employed by the Irish Department of Defence and he has published a similar book before <em>The Magic Numbers of the Professor</em> (2007), and published several papers on recreational mathematics. So he has had some training.</p>
<p>
The topics discussed are the classics: prime numbers, Fibonacci and Lucas sequences and Pascal's triangle, Pythagorean triples, triangular numbers, magic squares, the Monty Hall problem, and transcendental numbers: φ, π, e, √2, and the complex √-1. Most of these are also discussed in several other books. The lovers of this kind of books will recognize large parts, but there are some exceptions.</p>
<p>
Each of these topics is elaborated in a separate chapter. The keyword throughout the book is "patterns". The strategy is always the same. First some elements are written by the author and it always ends with comments and additions by Dr. Cong, a numerologist and obviously a friend of the author. O'Shea gives his biography at the end of chapter 2: a mathematical child prodigy originating from China, who immigrated to the USA. A dramatic ski accident prevented that he became a professional mathematician. Instead he participated in a traveling carnival where he took on the nickname Dr. Cong. The emails of Dr. Cong that are sometimes included use a style and assumes a level of readership that is suspiciously similar to what the O'Shea wrote, which blatantly confirms that he is just O'Shea's alter ego. Also in <em>The Magic Numbers of the Professor</em> the `Professor' is a fictitious character.</p>
<p>
The comments of Dr. Cong usually start by saying that the text is interesting but... and then he adds some extras to the topic discussed. His part is sometimes as long as the 'original text' that he is commenting on. However, he also adds (in my view very un-mathematical) numerological curiosities. For example the <em>lo shu</em> is an ancient Chinese 3 by 3 magic square that was exposed on the back of a turtle with the rows 4 9 2, 3 5 7, and 8 1 6. One of Cong's comments is that it refers to 666, the number of the beast because $4^3+9^2+2^1+8^3+1^2+6^1=666$. Or he comments in the third chapter that the first two letters of Pythagoras are the 16th and the 25th in the alphabet while the smallest Pythagorean triple is (9,16,25). A genuine mathematician's reaction to such statements would probably be: So what? And there are several other instances where dates, hours and other numbers can be combined to give so-called curious coincidences. In fact, there is a whole chapter on such "coincidences" which in my opinion diminishes the value of the book. These are indeed patterns, and it may attract extra readers who see tarot-like proofs in almost anything of whatever ethereal truth that is bestowed upon us by fate. It may (and should) shy away any readership of (potential) mathematicians. Of course it does make sense to explain that some coincidences are not as curious as one might think. For example the probability that two people in a group celebrate their birthday on the same day of the year is surprisingly high. That can be explained by simple statistics, but number fetishism is not mathematics. With this book, it is O'Shea's intention to make readers enthusiast for mathematics and take up interest in studying more of it. Solving a puzzle or finding out how a certain trick works may indeed be helpful. And there is indeed something to say about the recognition of patterns, but only if there is some rationality behind it that has to be discovered. But this kind of mysticism is a bad idea, or at least gives a very wrong impression of what mathematics is about. It is very easy to make such things up. For example, the figure in which Einstein's equation $E^2=(mc^2)^2+(pc)^2$ is represented on a Pythagoras triangle happens to be numbered 3.14, and moreover $3^1+4=7$, and 3.14 approximates π with 2 decimals in the fractional part, put 7 and 2 together and lo and behold, the figure appears on page 72. What a coincidence! But it is roaring nonsense.</p>
<p>
There are of course also quite nice things to say about this book. First of all there is its very elementary approach, but it is still discussing many mathematical objects and ideas. Sometimes they are only mentioned or briefly touched upon. Not really analytic proofs of course, but sometimes strong suggestions and indications are given for limiting value. Similarly we meet also Einstein's relativity theory as I mentioned above, but also several different geometric proofs of the Pythagoras theorem, continued fractions, Platonism, complex numbers, $i^i$, Schrödinger's equation, Stirling's asymptotic formula for the factorial, and many others.</p>
<p>
The author classifies his book under recreational mathematics. As he writes in the introduction, its purpose is to be entertaining and at the same time educational. He is obviously an admirer of Martin Gardner. But O'Shea's attitude towards mathematics is not as bad as I might have suggested above. To illustrate his vision on mathematics and how he sees his contributions in this book, I can quote what he writes on page 121. After it has been suggested (by numerical evidence) that the n-th root of the n-th Fibonacci number tends to φ (the golden ratio) as n tends to infinity, O'Shea writes "Of course there are those who ask what these curiosities tell us about our world. My answer to these questions is: They perhaps tell us nothing! Mathematics in itself does not <em>explain</em> our universe. Yes mathematics can be used in physics to explain how some parts of the universe operate. That is truly marvelous. But that is not why mathematics exists. Mathematics exists in its own right. It may well be the <em>only</em> reality. To find explanations on how the world works, I suggest one should study physics."</p>
<p>
There are a few of problems or puzzles to solve (but not many) which get solutions at the end of the chapter, and for the hungry reader there is a list of references to read more. Thus, if you are interested in mathematical issues and puzzles of the mathematical type, you will certainly enjoy this book even if you have only a minimal background. Just be aware that numerology is as alien to mathematics as penguins are to the Amazon jungle.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a book on recreational/popular mathematics. Not on magic and card tricks but with proper mathematical topics, Written at a really low level to be accessible for any layperson. There are however also a lot of patterns exposed that are of a numerological nature and these are in my opinion senseless observations in this context.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/owen-oshea" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Owen O'Shea</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-63388-148-8 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 19.00</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">330</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.prometheusbooks.com/index.php?main_page=product_info&amp;amp;products_id=2273" title="Link to web page">http://www.prometheusbooks.com/index.php?main_page=product_info&products_id=2273</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a08" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a08</a></li></ul></span>Sat, 07 May 2016 16:01:48 +0000Adhemar Bultheel46927 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/call-primes#commentsKepler and the Universe: How One Man Revolutionized Astronomy
https://euro-math-soc.eu/review/kepler-and-universe-how-one-man-revolutionized-astronomy
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Johannes Kepler (1571-1630) and Galileo Galileo (1564-1642) are the two names that pop up in most people's mind when it comes to the acceptance of the heliocentric planetary system that was previously announced by Nicolaus Copernicus (1473-1543). The ancient Greek tried to explain the trajectories of the Sun and the planets as circles with our Earth in the center. When that didn't fully explain things, Apollonius proposed epicircles and Ptolemy refined this a bit by placing the Earth just outside the center of the circles. That system was the general belief, until Copernicus picked up an idea of Aristarchus (3rd century BCE) and proposed a model where the Earth and the planets circled around the Sun. Some accepted this system not necessary as a reality but rather as a simplification for the computations to make predictions. There was also a compromise system proposed by the Danish astronomer Tycho Brahe where the planets were all circling around the Sun, but the Sun and the Moon were circling around a stationary Earth.</p>
<p>
Kepler was born in a divided Germany where the religion of a region depended on the choice of ruler of the region. This was an arrangement to bring some peace between Lutherans and Catholics. Kepler was Lutheran which is important to get his sacrilegious theories published. The heliocentric visions were considered unbiblical and were severely prosecuted by the Catholics as Galileo experienced in an Italy dominated by the Catholics. The Lutherans were a bit more tolerant, but still such theories were frowned upon also by them. Kepler had an unhappy youth in a wrecked family and he had many health problems. He writes in a later autobiography that many people seemed to hate him. He was a brilliant student and thanks to his mentor Michael Maestlin he got a position in Tübingen in pursuit of a clergy position. However, because of his Copernican ideas and Calvinist tendencies, he was expelled to a lower teaching position in Graz where he wrote his <em>Mysterium Cosmographicum</em>. He tries to answer three questions: why are there only 6 planets, why are they at a certain distance from the Sun, and why do more distant planets move more slowly? He (wrongly) thought to have found a solution for the first two questions by placing the five Platonic solids in between the planets. This shows that the ideas of the ancient Greeks were still strongly present in the Kepler's mind.</p>
<p>
The publication of this book brought him some fame and he got in contact with Galileo and Brahe. Especially the latter was important because Brahe had the most accurate observations available at that time. However Brahe was very protective about his data because he previously had a bad experience with another visitor, Nicolaus Reimer, who claimed Brahe's compromise solar system model as his own, resulting in a lifelong fight about plagiarism between these two. After Brahe died from benign prostatic hyperplasia, Kepler was appointed by emperor Rudolph II as a replacement for Brahe in Prague to work on the <em>Rudolphine Tables</em> that should allow to predict planetary positions. Now Kepler had access to Brahe's data and this led to his major work the <em>Astronomia Nova</em>. This is the work where Kepler formulates his famous second law, describing the speed of the planets on their elliptic trajectories around the Sun sweeping equal areas in equal time. Love gives a step by step analysis of this voluminous work of 600 pages. In this Prague period Kepler was very productive and wrote about optics, the human eye, observed a supernova that was named after him, he computed the precise date of Jesus' birth, and analysed the six-pointed snowflake. He described even space travel is a science fiction-like fashion.</p>
<p>
The first telescopes were produced which improved the accuracy of the observations considerably. Galileo published his <em>Sidereus Nuncius</em> (The Starry Messenger) based on telescope data which strongly supported the Copernican system. Kepler was very supportive of Galileo's work. His letter with commentaries was published as a book (Kepler was never concise) <em>Conversation with Galileo's Messenger from the Stars</em>. The discovery of four moons of Jupiter by Galileo led Kepler to the prediction that Mars should have two moons and Saturn six or eight. This was based on a wrong arithmetic or geometric progression that he thought to be present in our solar system. On the other hand he formulated several other speculations that turned out to be rather close to reality. For example his argument used to explain Olbers' paradox: the fact that it is dark at night can be explained if we accept a finite universe. Galileo announced more of his findings in the form of anagrams (to claim later priority since scientific journals did not exist in those days). Some of these were wrongly deciphered by Kepler. Such misinterpretations of facts seemed to have happened frequently to Kepler. He sometimes stuck to a wrong idea and predicted something based on wrong arguments that later turned out to be surprisingly accurate.</p>
<p>
Kepler's third law was written up in his book <em>Harmonices Mundi</em> that he wrote while he was in Linz. This law relates the distance of a planet from the Sun with the period of its revolution. This finally settled the answer to the third question he had when writing his <em>Mysterium Cosmographicum</em>. After that he eventually finished and published the <em>Rudolphine Tables</em> during the last period of his life that he spent in Ulm.</p>
<p>
All this work he did notwithstanding all the misfortune he had in his personal life. He married and was widowed twice. He had twelve children of which only three reached adulthood. He had problems with the heirs of Brahe, and he had to watch his steps on the thin ice of religion and politics. His mother died shortly after she was acquitted of witchcraft and his life ended in the middle of the Thirty Years War that eliminated one third of the German population. He died probably in a bout of quartan fever of which he suffered intermittently most of his life.</p>
<p>
Love brings this biography in a very readable form, also for somebody who is not a historian or a specialist in celestial mechanics. He is obviously an admirer of Kepler, but does not shy away to expose where Kepler has erred. The main thread of the book is not Kepler's personal life, but the subdivision in chapters and the chronology is directed by Kepler's scientific results that eventually have led to his three laws of celestial mechanics. There are many illustrations and at the end an epilogue is added prolonging the post-Keplerian history of astronomy and cosmology via Newton and the eighteenth till the twenty-first century. This allows Love to mention the Kepler crater on the moon and NASA's Kepler mission launched in 2009 to discover earth-like planets in our universe.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a very readable biography of Johannes Kepler with special attention on how he obtained his three laws in celestial mechanics. Therefore it is also an historical account of an exciting period in astronomy in the second half of the sixteenth and the first half of the seventeenth century.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/david-k-love" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David K. Love</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-63388-106-8 (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">253</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.prometheusbooks.com/index.php?main_page=product_info&amp;amp;products_id=22" title="Link to web page">http://www.prometheusbooks.com/index.php?main_page=product_info&products_id=22</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-01</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/70a15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">70A15</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/85-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">85-03</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a44" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A44</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/01a45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A45</a></li></ul></span>Fri, 27 Nov 2015 16:51:39 +0000Adhemar Bultheel46569 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/kepler-and-universe-how-one-man-revolutionized-astronomy#commentsNumbers: Their Tales, Types, and Treasures
https://euro-math-soc.eu/review/numbers-their-tales-types-and-treasures
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Numbers and (elementary) number theory is often used in books about popular mathematics since many problems there are easy to formulate at a generally understandable level. Moreover the history of numbers and their notation goes along with the history and the evolution of mathematics. So we see that numbers feature in books like <a href="/review/single-digits-praise-small-numbers" target="_blank">Single digits</a> (M. Chamberlain, 2015), <a href="/review/professor-stewarts-incredible-numbers" target="_blank">Professor Stewart's incredible numbers</a> (I. Stewart, 2015), <a href="/review/those-fascinating-numbers" target="_blank">Those Fascinating Numbers</a> (JM De Koninck, 2009), <a href="/review/magic-numbers-professor" target="_blank">The Magic Numbers of the Professor</a> (O. O'Shea, 2007), <a href="/review/numbers-work-cultural-perspective" target="_blank">Numbers at Work </a> (R. Taschner, 2007), and this list goes on and on, and then there are the many books on the history of numbers like <em>The Universal History of Numbers</em> by G. Ifrah (Wiley, 2000).</p>
<p>
And here is yet another popular book about numbers. We do find some elements that can also be found elsewhere, but there are still some differences. The book starts in its first chapter with the obligatory association of numbers with counting and what counting really means, which is conceptually not as trivial as it seems since to do computations, you have to detach numbers from the objects you are counting. Only then the abstract notation of numbers is initiated. A more detailed history of the notation of numbers is given in chapter 3. The interlude of chapter 2 elaborates on the psychological aspects of numbers. We instinctively grasp small amounts up to three or four, which is reflected in the separate, non-systematic names that are given to these numbers in practically all languages, even the most primitive ones. If there are more than 5 objects, we start counting or catch their number by grasping couples or triples and calculate. Counting is not intuitive. We have to learn it by practicing. A child that can name the numbers from one to ten is reciting a poem, which is quite different from counting.</p>
<p>
Once the abstraction from counting objects to numbers is properly made, one can start detecting patterns like triangular, square, or rectangular numbers, meaning that we can arrange that amount of stones or objects or dots in the form of a triangle, square, or rectangle. These arrangements can lead to well known summation formulas. Pentagonal and tetrahedral numbers are less common and certainly less popular.</p>
<p>
The fifth chapter is called <em>Counting for Poets</em> and may come a bit as a surprise in a book on numbers. However certain patterns of metrical rhythm are imposed by the type of the poem. It may require stressing syllables by duration or loudness. Verse meters are as old as the Vedic literature. The duration or weight of a verse is expressed in units of moras. For example a meter is a sequence of short and long syllables, of weight respectively one and two moras. One may see here an analogy with a music score. So, given a the total length of the verse, how many different meters are then possible? And so, triggered by Pingala's historical work on Sanskrit prosody, this is how the authors turn this into a chapter on numbers and combinatorial problems. Even the Fibonacci sequence shows up. The Fibonacci's sequence, the golden section, and Pascal's triangle are further discussed in the next chapter.</p>
<p>
With several patterns that can be detected in Pascal's triangle, the kickoff is given for other arrangements of numbers like magic squares and how to construct them. Napier's rods or bones are sticks with numbers written on it, and when arranged in an appropriate way, this can be used to multiply (moderately) large numbers by recognizing certain patters. It is basically an Arabic invention, but it was improved by Napier who is better known for his invention of the logarithms. You may read more about this in <a href="/review/john-napier-life-logarithms-and-legacy" target="_blank"><em>John Napier: Life, Logarithms, and Legacy</em></a> (John Havil, 2014).</p>
<p>
Chapter 8 finally arrives at the unavoidable prime numbers, but this is kept rather short, culminating in a list of unsolved problems. Other special numbers are perfect numbers, Kaprekar numbers, Armstrong numbers and some isolated numbers with surprising properties.<br />
Relationships between numbers are the subject of the next chapter like amicable numbers of all sorts. Most attention goes however to Pythagorean triples, their construction and some curious properties and patterns they reveal. Also checks for divisibility by 2,3,...,17 are explored.<br />
The keyword for chapter 10 is proportions. Continued fractions are introduced for example when subdividing an interval with the ratio of Fibonacci numbers, which of course results in the golden ratio (again), but also pi as the ratio of the circumference of the circle to its diameter and its long history are the subject here.<br />
The last chapter is about numbers and philosophy as in the 20th century numbers were defined in a formal way by logicists, formalists as well as constructivist. You will find here also some thoughts about the modeling of nature, the so called <em>Unreasonable effectiveness of mathematics in the natural sciences</em> as Wigner once stated it.<br />
Some tables of numbers (Fibonacci, primes, perfect numbers, Kaprekar numbers, Armstrong numbers, amical numbers, and palindromic numbers) conclude the book.</p>
<p>
This survey illustrates that there are some well known topics discussed, but also some unexpected ones, that makes this book certainly advisable. No special mathematical training is needed. Especially the chapters on the development of the concept of numbers in a child and the link with prosody are refreshing. Some of the intriguing patterns presented are unexpected and new to me. So I can full-heartedly recommend to read this book both for the professional mathematician and the generally interested reader.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Obviously a book about numbers. Some of it is mathematics as in elementary number theory: of course primes and other interesting numbers and intriguing number patterns that can be observed in all kinds of tables. But you also learn about the history of these numbers, the psychological aspects of counting and philosophical aspects in defining numbers. Somewhat surprisingly, you also learn about the link between prosody and numbers and how patterns of verse meters in turn lead to combinatorial problems.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/alfred-s-posamentier" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">alfred s. posamentier</a></li><li class="vocabulary-links field-item odd"><a href="/author/bernd-thaller" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Bernd Thaller</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-63388-030-6 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">19,00 USD</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">400</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.prometheusbooks.com/index.php?main_page=product_info&amp;amp;products_id=2253" title="Link to web page">http://www.prometheusbooks.com/index.php?main_page=product_info&products_id=2253</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a05</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/11-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11-01</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/97f60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97F60</a></li></ul></span>Mon, 17 Aug 2015 10:22:06 +0000Adhemar Bultheel46361 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/numbers-their-tales-types-and-treasures#commentsGreat calculations: A surprising look behind 50 scientific inquiries
https://euro-math-soc.eu/review/great-calculations-surprising-look-behind-50-scientific-inquiries
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
Many books have been written that are popularizing mathematics and science in general. The history is rich of anecdotes, oddities, and amazing facts so that one may easily fill up an entertaining book with these. The approach that Colin Pask has taken here is however original. Science started because people wanted to find answers to simple questions about the world they live in or about themselves. The progression of science depends on an interplay of theory and experiment, leading to models that are gradually modified and refined. Sometimes experiments detect anomalies that force the model to be changed, sometimes, the model predicts things that need to be verified by experiments. Whether it are experiments or theoretical predictions that are at the origin, there are always some calculations involved. So it makes sense to place these calculations at the center of major advances made in science. Selecting only 50 of them must be a big challenge, but this is what Pask has done, although he apologizes for the many others he has left out and that may have been closer to the heart of some readers.</p>
<p>
When I say science, in this context, I mean the most formalized science. In this book, that is besides mathematics, mostly physics and astronomy and a bit of life-science. In this computerized age, it is difficult to imagine that these calculations were achieved with the tools available at that time. Think of the first estimation of pi by Archimedes. Taking into account that Fibonacci's <em>Liber Abbaci</em> popularized the Hindu-Arabic numerals only in 1202, then the richness of data in the <em>Almagest</em> of Ptolemy, or Kepler's astronomical computations are mindblowing achievements. And there are many obvious questions but with nontrivial answers that people have found answers for in the past. Why is the sky dark at night? How large and how old is the Earth? Why does the sun shine? Etc. And there are the more recent achievements in particle physics and cosmology. Just think of your own favorite major event in the history of science and you will notice that there is always some major calculation involved. Sometimes it is the number that is the answer, but sometimes it are not the numbers, but the pattern they represent that one is looking for.</p>
<p>
Pask has organized his top 50 calculations in coherent chapters. Within the chapters, some historical chronology is maintained, as is loosely speaking also the case for the succession of the chapters. The first chapters are about mathematics (Pythagorean triples, logarithms, pi, prime numbers,...). The next chapters collect topics concerning our planet earth (its age, size, mass, tides,...), the solar system (heliocentric model, why the moon stays in orbit, detection of the planets, Halley's comet,...) and the universe (dark sky, its topology, the origin of chemical elements, dark matter, escape velocity,...). Then there is a chapter on life-science topics (blood circulation, population dynamics, annuity pricing, genetics, computer tomography, scaling for living species,...). The remaining chapters deal with physics. First there is optics (the speed of light, the colours of the rainbow, waves, electro-magnetism, photons, and relativity). Then we get the building blocks of our universe from atoms and Brownian motion to quantum and particle physics, fusion and fission. Dynamical systems is the subject of the penultimate chapter starting with Fourier analysis over Bessel functions to nonlinear dynamics and chaos.</p>
<p>
From this skimmy survey it should be clear that the subjects are quite diverse and one can imagine that not all the mathematics and calculations are equally accessible to a broad audience that Pask is obviously addressing. So he has chosen to give for the simplest ones all the details, but not for others that are more involved. However, even if it becomes a bit technical, one van easily skip the details and get the general picture. And he always has some entertaining story to tell, and sometimes also he has to tell something `behind the scene'. It is interesting to read how scientists working on the same problem had different views. Pask gives many original citations and he gives detailed references in case one wants to know more about an event. Not so much about the mathematics or the technical content, but often advise is given like `an excellent introduction to the topic is...' `read all about the story behind this in...', 'my favorite book on this topic is...', 'the recent book ... uses pictures to illustrate...' etc. Even professional scientists will discover something they did not know. For example, G.H. Hardy, known for his praise of pure mathematics and thinking low of any application it may have, had chosen number theory as his main topic expecting it to be most detached from applications. However, we know that prime numbers are now essential for cryptography, but what I did not know was that his name is also attached to a well known Hardy-Weinberg law in genetics because of his 1908 paper in <em>Science</em>. Or this one: Heisenberg (deliberately or not) largely overestimated the amount of uranium needed to make the atomic bomb, which made Germany decide it was not feasible to proceed with the production of such a bomb. Or that John Adams had predicted the position of Neptune but how Leverrier and Galle got the glory. That the FPU project (named after Fermi, Pasta, and Ulam) on one of the first computers at Los Alamos to simulate the vibration of a string was programmed by Mary Tsingou, a major achievement in those days of emerging computers, but she never got proper recognition for it. Pask also apologizes that only three women are involved in his 50 calculations although during WW II just before computers were invented, the human computers were mostly female since they had the reputation of being more careful. Besides the many references and notes listed by chapter at the end, there are also many illustrations. Some are reproductions from the publications and others are produced by Annabelle Boag.</p>
<p>
It is only in his last chapter that Pask explains his criteria for selecting his 50 favorite calculations, and then he continues by considering this as the shortlist for selecting his top 10. Depending on the criteria used and the personal preferences, this top list will differ for every reader. There will probably be overlap, but I am sure there will be many others that were not even mentioned in this book. So, since many relevant calculations are left untold, I am already looking forward to subsequent volumes with additional calculations. The questions asked (and answered) in the book, the level of exposition, the readability, the pleasant style, and the decorating stories make this book highly recommendable for anyone interested in mathematics and science and its history.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The author gives a list of 50 historical favorite that had a great impact on the history of science. His discussion of these is accessible for a broad public that is interested in mathematics, science and its history. The topics include mathematics, but mainly physics, including astronomy, cosmology and particle physics, and to a lesser extent life sciences.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/colin-pask" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Colin Pask</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/prometheus-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">prometheus books</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-1-63388-028-3 (pbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">USD 18.00</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">380</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.prometheusbooks.com/index.php?main_page=product_info&amp;amp;cPath=60&amp;amp;products_id=2249" title="Link to web page">http://www.prometheusbooks.com/index.php?main_page=product_info&cPath=60&products_id=2249</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a09</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a05" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a05</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a72" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A72</a></li></ul></span>Mon, 20 Jul 2015 16:01:51 +0000Adhemar Bultheel46313 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/great-calculations-surprising-look-behind-50-scientific-inquiries#comments