European Mathematical Society - 01A50
https://euro-math-soc.eu/msc-full/01a50
enEuler's Gem
https://euro-math-soc.eu/review/eulers-gem-0
<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 fact that the book is reprinted in its original version as a volume of the <em>Princeton Science Library</em> is a quality label as such. For completeness, I should here also mention the subtitle: "The polyhedron formula and the birth of topology". This rules out that by the "gem" is not meant the other famous Euler formula $e^{i\pi}+1=0$, but that it concerns the polyhedral formula <em>V</em>−<em>E</em>+<em>F</em>=2 (the letters stand for Vertices, Edges and Faces).</p>
<p>
A short introduction serves as a teaser for the reader and explains the kind of problems that will be discussed in the sequel. A reasonable way to start the full story is to give a biography of Euler. He is recognized as one of the greatest mathematical minds of all times. Next, the reader is surprised by the fact that it is not so straightforward to define a polyhedron, or at least to identify the ones that one wants to focus on. So the reader is warped back to the Greek origin when the five Platonic solids (discussed in Book XIII of Euclid's Elements) and the thirteen Archimedean solids were studied. We then have to take a leap to the Renaissance of the fifteenth century before the polyhedra and the Greek knowledge was rediscovered. Kepler later built a whole world view and a solar system on polyhedral shapes. And then came Euler, who detected his above mentioned "gem". It is so simple an observation that it comes as a real surprise that, as far as we know, it was missed by everyone so far. Some explanation may be that previously one concentrated on the vertices and the faces (or the solid angles), but Euler also considered the edges as essential components of a polyhedron. Richeson reproduces Euler's proof (1750-51), but the formula does not hold for all (regular) polyhedra. When does it hold and when does it not? The reader is expecting to read the answer in the next chapter, but then Richeson surprises again by revealing that Descartes may have been the first one to have discovered the formula because a similar relation was found posthumously in his notes (notes that were miraculously saved from oblivion). However, a complete rigorous proof was only given by Legendre (in 1794), a proof that Richeson also explains later in the book.</p>
<p>
From the story told so far, there are already many historical mathematicians involved and Richeson gives every time some short biographical sketch to situate him (so far only men) as a person who existed and lived a life of his own. It is not just some abstract name used to identify a result. Note also the way in which Richeson builds up his story. He takes the reader along to think about what polyhedra are, for what polyhedra does the formula hold, and how could it be adopted to hold in more general case? Euler and his proof are some kind of a climax, but then Descartes shows up as an unexpected twist, and Legendre's proof is not based on properties of planar faces but (another surprise) requires geometry on a sphere with geodesics. This shows how well the book is written and how Richeson manages to fascinate the reader, and make her curious about what is coming up next.</p>
<p>
And next chapter is again some kind of a surprise because it introduces the problem of the Bridges of Köningsburg and how Euler solved the problem which is considered as the origin of graph theory. Not so surprising though if you know that a graph consists of vertices and edges. Cauchy uses this idea by projecting a polyhedron on a plane, giving a plane graph that can be analysed to prove Euler's formula. Now Richeson's story takes off into graph theory and applications: recreational mathematics (the game of sprouts and Brussels sprouts invented by Conway), the four colour problem for planar maps (and other graph colouring problems). Graph based proofs for the polyhedra formula and generalisations concludes the graph theory subject.</p>
<p>
Next Richeson embarks on proper topology as the rubber sheet version of the usual geometry. This requires some new concepts and a classification of all surfaces. Therefore one needs to know when a surface is or is not homeomorphic to another and thus are topologically the same. For example, a torus is a sphere with a handle, a Möbius band is the same as a cross cap, and the projective plane is a sphere with a cross cap. Classification is connected to the definition of the Euler characteristic (or Euler number as Richeson calls it). Make a finite partition of the surface and count the "rubber versions" of vertices, edges and faces, then the Euler formula gives the characteristic <em>χ</em></p>
<p>
which is an invariant for the surface (2−2<em>g</em> for a sphere with <em>g</em> handles, and 2−<em>c</em> for a sphere with <em>c</em></p>
<p>
cross caps). This characteristic and the orientability of the surface allows some classification as started by Riemann but only completed in 1907 by Dehn and Heegaard.</p>
<p>
I have to say that, although now Richeson is still explaining things at an introductory level of topology, (and continues to do so), it will take a more persistent and motivated topological layman to follow in pace and read on. We arrived now at about two thirds of the main text of the book and the mathematical level is not decreasing for the last part. It continues with knots and Seifert surfaces (whose boundary is a knot or link), the hairy ball theorem for vector fields on a sphere and more generally the Poincaré-Hopf theorem on surfaces with boundary, Brouwer fixed point theorem, the angle excess theorem for a surface, the Gauss-Bonnet theorems about the total curvature of an orientable surface, Betti numbers, and Richeson ends with an epilogue about the Poincaré conjecture. All of these are nicely presented in a smooth and logical succession by Richeson, but they are too technical to be discussed at the level of this review. However, for example an undergraduate mathematics student should not have a serious problem to read on.</p>
<p>
Everything in the book is very well illustrated with insightful graphics that, together with the text, make results almost like being obvious. Richeson also adds in an appendix building patterns that can be used to make paper models of polyhedra, of a (square and edgy, yet topologically a perfect) torus and even (the paper realization that looks like) a Klein bottle, or a projective plane. In the text he also gives advise on how to prepare the liquid to make soap-bubble models. These are aids to help visualising the surfaces if the graphics of the text do not suffice. There is a long list of papers referred to in the text, but also an appendix with an annotated survey of recommended literature.</p>
<p>
Except for an additional preface by the author, the book is the unaltered reprint of the original version of 2009. Thus for example the facts of Martin Gardner passing away in 2010 and Perelman refusing the Millennium Prize for proving the Poincaré conjecture were still unknown in 2009. Although the latter was to be expected since he had already declined the Fields Medal in 2006 and an EMS Prize.</p>
<p>
The first half of the book can be considered as a popular science book on a mathematical subject written for everyone. Depending on the motivation or knowledge of the reader this might or might not include the part on graph theory. Once Richeson dives deeper into topology, it becomes more a popular science book for the mathematics student of at least an amateur mathematician. People who are interested in this book may also be interested in a more recent book by Richeson that has also been reviewed here <a href="/review/tales-impossibility" target="_blank"><em>Tales of Impossibility</em></a>.</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 book of 2009, that is now reprinted in the <em>Princeton Science Library</em>. Richeson gives an account of 2500 year of mathematical history that runs from the Greek's approach to regular polyhedra to the modern problems of topology, all centred around Euler's polyhedral formula <em>V</em>−<em>E</em>+<em>F</em>=2 and its generalisations.</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-s-richeson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David S. Richeson</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/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</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">2019</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">9780691191379 (pbk), 9780691191997 (ebk)</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">£ 14.99 (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">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/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li><li class="vocabulary-links field-item odd"><a href="/imu/topology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Topology</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://press.princeton.edu/books/paperback/9780691191379/eulers-gem" title="Link to web page">https://press.princeton.edu/books/paperback/9780691191379/eulers-gem</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/55-algebraic-topology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">55 Algebraic topology</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/55-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">55-03</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/52-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52-03</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/54-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">54-03</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/01a50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A50</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/51m20" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51m20</a></li></ul></span>Fri, 31 Jan 2020 10:49:42 +0000Adhemar Bultheel50363 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/eulers-gem-0#commentsEuler's pioneering equation
https://euro-math-soc.eu/review/eulers-pioneering-equation
<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>Since The Mathematical Intelligencer conducted a poll in 1988 about which was the most beautiful among twenty-four theorems. Euler's equation $e^{i\pi}+1=0$ or $e^{iπ}=−1$ turned out to be the winner, and that is still today largely accepted among mathematicians. Even among physicists this is true. In a similar poll from 2004, it came out second after Maxwell's equations. The subtitle of this book is therefore <em>The most beautiful theorem in mathematics</em>.</p>
<p>This may immediately raise some controversy, not about the choice of the formula, but perhaps about what it should be called: a theorem, an identity, an equality, a formula, an equation,... A theorem or a formula applies but these are quite general terms. The others refer to formulas with an equal sign. The term identity assumes that there is a variable involved and that the formula holds whatever the value of that variable. That applies to Euler's identity, which is the related formula $e^{ix}=\cos(x)+i\sin(x)$. The previous formula appears as a special case of this identity. Wilson calls the former formula and "equation" but the reader with some affinity to the French language would probably prefer to call it an equality because the French équation means it has to be solved for an unknown variable. But all the previous names have been used interchangeably to indicate the formula. Calling it <em>Euler's identity</em> may not be the most correct but it is probably the most common terminology.</p>
<p>Whatever it is called, the description, if not <em>most beautiful</em>, then certainly the qualification <em>most important</em> or <em>most remarkable</em>, would be well deserved. It involves five fundamental mathematical constants: 1,0,π,e, and i in one simple relation. The 1 generates the counting numbers. The zero took a while to be accepted as a number but also negative numbers were initially considered to be exotic. Rational numbers were showing up naturally in computations, but so did numbers like √2 and π. These required an extension of the rationals with algebraic irrationals like √2 and the transcendentals like π which results in the reals that include all of them. The constant e (notation by Euler) relates to logarithms and its inverse the exponential function growing faster than any polynomial. Finally the imaginary constant i = √-1 (which is another notation introduced by Euler) was needed to solve any quadratic equation. This i allowed to introduce the complex numbers so that the fundamental theorem of algebra could be proved. The exponential and complex exponential are essential in applied mathematics. Euler's identity is most remarkable because it relates exponential growth or decay of the real exponential, and the oscillating behaviour of sines and cosines in the complex case.</p>
<p>All these links allow Wilson to tell many stories about mathematics that are usually discussed in books popularizing mathematics for the lay reader. There are indeed five chapters whose titles are the five previous constants and a sixth one is about Euler's equation. He does this in a concise way. The amount of information compressed in only 150 pages is amazing. This doesn't mean that it is so dense that it becomes unreadable. Quite the opposite. Because there are no long drawn-out detours, the story becomes straightforward and understandable. For example the first chapter (only 17 pages including illustrations) introduces children's counting rhymes, compares the names for numbers in seven different languages, and compares number systems: Roman, Egyptian, Mesopotamian, Greek, Chinese, Mayan, and the Hindu-Arabic. The latter was popularized in the West by Fibonacci and Pacioli. There are many illustrations not only of the notation of these different numerals in this chapter, but there are in fact many other illustrations throughout the book. This does not increase the number of pages needlessly because a picture sometimes says more than a thousand words. There are no colour illustrations but colour is not relevant for what they represent.</p>
<p>This is not the first book on Euler's equation. For example Paul Nahin. <a target="_blank" href="/review/dr-eulers-fabulous-formula"><em>Dr. Euler's Fabulous Formula</em></a>, Princeton University Press (2006), which is a bit more mathematically advanced, and a more recent one by David Stipp. <em>A Most Elegant Equation</em>, Basic Books (2017), which has more info about the person Euler. In the current book Euler's name appears frequently but as a person he is largely absent. For most of the five constants, separate popularizing books have been written or they are discussed in a chapter of more general popular books about mathematics, too many to list them here. Wilson refers to some of them in an appendix with a short list of additional literature, conveniently listed by subject.</p>
<p>There is of course mathematics in this book. It would be weird if there wasn't. But there is nothing that should shy away a reader with a slight affinity for mathematics. Some of it can be skipped, but the exponential and trigonometric functions, series, and an occasional integral do appear. The more advanced definitions or computations, are put in one of the eleven grey-shaded boxes distributed throughout the book, so that skipping is easy. Most of the topics are placed in their historical context. For example, the history of the computation of π is well represented, and also the history of the logarithms as they were derived by Napier and Briggs and how they relate is nicely explained. There are some notes to explain how complex numbers can be generalised to quaternions and even octonions, and several examples from applied mathematics illustrate the meaning and relevance of the exponential function.</p>
<p>A minor glitch: Albert Girard (1595-1632) who was the first to have formulated the fundamental theorem of algebra, is called on page 116 a Flemish mathematician, which is strange because the man was born in France, but, as a religious refugee, moved to Leiden in what was then the Dutch Republic of the Netherlands. So I do not think that the characterization Flemish does apply here.</p>
<p>The book does not go deep into the subjects discussed, but I liked it because it is quite broad, touching upon so many mathematical subjects, mainly in their historical context, while readability remains most enjoyable notwithstanding its conciseness.</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 in which Wilson gives a popularizing account about the historical development of mathematics. His guidance is Euler's equality that connects five fundamental constants of mathematics: 1, 0, π, e, and i = √-1. Each of these is an incentive to discuss respectively different number systems, how counting extends to negative numbers and eventually the real numbers, the approximation and calculation of π, different logarithms, and complex numbers.</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/robin-wilson" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robin Wilson</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/oxford-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">oxford university press</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">2018</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">9780198794929 (hbk); 9780198794936 (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">£ 14.99 (hbk); £ 9.99 (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">176</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://global.oup.com/academic/product/eulers-pioneering-equation-9780198794936" title="Link to web page">https://global.oup.com/academic/product/eulers-pioneering-equation-9780198794936</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/97a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">97A80</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A50</a></li></ul></span>Sun, 14 Apr 2019 07:22:45 +0000Adhemar Bultheel49288 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/eulers-pioneering-equation#commentsAn Imaginary Tale: The Story of √-1
https://euro-math-soc.eu/review/imaginary-tale-story-%E2%88%9A-1
<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 a reprint in the <em>New Princeton Science Library</em> of a classic. The series brings reprints in cheap paperback and eBook format of classics, written by major scientists and makes them available for a new generation of the broad public. The series includes not only math books but covers a broader area, although there are several mathematics classics in the catalog written by E. Maor, J. Havil, and R. Rucker, but also J. Napier, A. Einstein, O. Toeplitz, R. Feynman, S. Hawking, R. Penrose, W. Heisenberg, etc. So if you missed out on some of the original editions, or were not even born at that time, this is a chance to get one of these more recent reprints. Another classic, reprinted at about the same time is Maor's <a href="/review/e-story-number"><em>e: the story of a number</em></a> which follows a similar idea that may have inspired the current author.</p>
<p>
The current reprint is of the first paperback edition of 2007 which is an updated version of the original from 1998. Paul Nahin is an electrical engineer who wrote several successful popular science books. His first one was a biography of Heaviside, and this book about complex numbers (it contains even an introduction to complex functions) was his second. Several other were to follow, some of which have been reviewed in this EMS database: <a href="/review/chases-and-escapes-mathematics-pursuit-and-evasion"><em>Chases and Escapes. The Mathematics of Pursuit and Evasion</em></a> (2007), <a href="/review/digital-dice-computational-solutions-practical-probability-problems"><em>Digital Dice. Computational Solutions to Practical Probability Problems</em></a> (2008), <a href="/review/number-crunching"><em>Number Crunching</em></a> (2011), <a href="/review/logician-and-engineer-how-george-boole-and-claude-shannon-created-information-age"><em>The Logician and the Engineer. How George Boole and Claude Shannon Created the Information Age</em></a> (2012), <a href="/review/holy-sci-fi-where-science-fiction-and-religion-intersect"><em>Holy Sci-Fi! Where Science Fiction and Religion Intersect</em></a> (2014), Of course complex numbers and functions are important tools in electrical engineering. The book has a strong historical component of course, but, unlike the book by Maor about the history of the number e, this book has much more mathematics in it. Hence it requires some mathematical affinity to understand much of what is presented here. It requires the knowledge of advanced secondary school or even freshman's university level, in particular when it turns into an introductory course on complex functions in the trailing chapter. Nevertheless, Nahin avoids a textbook structure of definitions, theorems and proofs, but keeps the level of a casual account, cheering up the reader with some witty remarks now and then.</p>
<p>
The historical background starts with the solution of the cubic equation and the search for a formula that gives its roots, which was a hot topic in the 16th century. Knowing such a formula was a strong weapon for `mathematicians' that made a living as (human) computers, so it was important not to share it with competitors. Obviously this includes the well known story of del Ferro who knew how to solve the cubics and who told it to Antonio Fior before he died. Niccolo Fontana, better known as Tartaglia, also knew how to solve them. It came to a public duel between the latter two to solve the most equations in a given time. Tartaglia won much to Fior's surprise. Cardano who was a well respected mathematician in those days, stalked Tartaglia to tell him the magic formula, and after much pressure, Tartaglia eventually told Cardano, but made him promise to keep it a secret. However Cardano could consult the letters by del Ferro and considered his promise to Tartaglia not valid anymore and published the result anyway, which resulted in a vigorous priority fight.</p>
<p>
Solving cubic equations is important for the history of complex numbers because the square roots in the relatively complex formulas could give complex conjugate solutions. However, even though the square roots of negative numbers made no sense to them, it turned out that when computations were performed as if these were genuine numbers, this could lead to valid results, which was most puzzling at first. The formulas are known as Cardano's but historically this is clearly a mistake. They were re-discovered a couple of times by others (e.g. Leibniz).</p>
<p>
Since in antiquity (think of the Greek) many computations were done by geometric constructions. Descartes, Wallis, Newton, and others thought about a construction of the square root with compass and ruler. In geometric constructions it is difficult to give a meaning to a negative number when it concerns the length of a line segment or the area of a polygon. It was not before Bombelli had the idea of drawing a line with marks for the numbers (which we now call the real line) that negative numbers finally made sense. It was even more staggering in those days to make geometric sense of an imaginary number. Some possible interpretation was that a line intersected another one outside an certain interval, which made the intersection `imaginary'.</p>
<p>
Of course, as we know, the proper interpretation of a complex number should be a point in the complex plane. That idea and the modulus-argument representation of the complex numbers with a complex exponential was first given by the Norwegian Caspar Wessel in 1797. He was not even a professional mathematician and succeeded where many great minds had failed before him. His finding went unnoticed though, until much later. Argand and Buée came to the same solution about a decade later and published their findings almost simultaneously which started another row between them. Again these names do not sound very familiar to us. Argand was a French amateur mathematician and Buée was a French priest who published a confusing, almost mystical paper on the subject. Just like Wessel's, their results faded away and were only re-discovered much later. Of course once the complex plane is accepted, one gets the goniometric representation, the formulas of De Moivre, the multiplication with i corresponds to a rotation over 90 degrees, etc. This is close to the vector interpretation by Hamilton, who finally gave a formal definition of the complex number field where a complex number was a couple of reals with vector addition and scaling and with a particular way of multiplying the vectors.</p>
<p>
Once the complex playground has been fixed in the first three chapters, the next three chapters deal with applications of complex numbers. Since the complex numbers are like vectors in a plane, some geometric problems can be easily solved with complex arithmetic. A less known theorem of Cotes and a puzzle problem from Gamow's book <em>One, Two, Three... Infinity</em> are given as examples. Other applications discussed include the `imaginary' time axis in the space-time indefinite metric of relativity theory, the maximal distance of a random walk with decreasing step sizes, Kepler's laws, and electrical circuits. More mathematics is found in the chapter on Euler and the famous Euler formula (exp(ix) + 1 = 0) but also infinite series, infinite products, the calculation of i to the power i, and even the gamma and zeta functions. The final chapter is an introduction to complex functions, derivatives, contour integration, Cauchy integral theorem, Green's theorem, analytic and harmonic functions.</p>
<p>
Concerning the structure of the text, I can mention that it is occasionally interrupted by `boxes' that discuss some topic closely related to the surrounding text, but that is not essential and could be skipped without a problem. On the other hand, some technical material is moved to appendices. The reprint is the unaltered version of 2007. That means that no additions or corrections are added since and the little defects that remained are still there. For example, the strange looking capital gamma symbol on page 177 (it is at least a different font from the surrounding pages) is still there. The preface of 2007 does explain what corrections and additions were made on that occasion. This book is not recommended if you are allergic to formulas, but if you want to peek behind the formulas and theorems in a textbook (a textbook is more `to the point' and hence necessarily `duller'), this is a the book that I recommend to read. You will definitely enjoy it. In fact it clearly reflects the the joy and delight that the author experienced when he was confronted with complex analysis during his engineering studies. </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 paperback reprint in the New Princeton Science Library of the bestselling original from 1998. It tells the story of the square root of –1, and that includes complex numbers, how they came about and what they can be used for. At the end there is even a brief introduction to complex functions. </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/paul-j-nahin" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">paul j. nahin</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/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</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">9780691169248 (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">£11.95</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">296</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="http://press.princeton.edu/titles/9259.html" title="Link to web page">http://press.princeton.edu/titles/9259.html</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/01a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A40</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A45</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/01a50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A50</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/30-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">30-03</a></li></ul></span>Wed, 27 Apr 2016 09:27:50 +0000Adhemar Bultheel46900 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/imaginary-tale-story-%E2%88%9A-1#commentsLeonhard Euler: Mathematical Genius in the Enlightenment
https://euro-math-soc.eu/review/leonhard-euler-mathematical-genius-enlightenment
<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>
Leonhard Euler (1707-1783) is not only the prominent mathematician of the 18th century, he was also physicist, astronomer, engineer, and administrator. He contributed to mathematics in analysis, graph theory, number theory, geometry and many others, but he also was quite active in mechanics, fluid dynamics, optics, music theory, and astronomical observations. Many papers and books were devoted to his life and work already. Some were written, revived or translated on the occasion of the tercentenary year 2007 which brought us an English translation of E. Fellmann's German biography from 1995 and even a graphical novel by Heyne and Heyne (see e.g. <a href="/review/leonard-euler" target="_blank">here</a> and <a href="/review/leonhard-euler-ein-mann-mit-dem-man-rechnen-kann" target="_blank">here</a> for a review). His biography in French by <a href="http://www.medhyg.ch/index.php/catalog/product/view/id/1378/%28language%29/fre-FR" target="_blank">Ph. Henry</a>, put emphasizing on Euler as a geometer, and there are of course the <a href="http://www.maa.org/press/periodicals/convergence/euler-tercentenary-volumes" target="_blank">MAA tercentenary volumes</a> and there are more. But besides the books about Euler, there is the enormous output of Euler himself, being collected in more than eighty volumes of the <em>Opera omnia</em>. Many of his papers are available in electronic form at the <a href="http://eulerarchive.maa.org/" target="_blank">euler archive</a> of the MAA and some were discussed in E. Sandifer's monthly column <a href="http://eulerarchive.maa.org/hedi/" target="_blank">How Euler did it</a>. Of all the biographical publications, the present book is the most complete from an historical perspective. Both Euler's life and his work is discussed, but the the technicalities of the mathematics as such, like for example the discussions of Sandifer's columns, are not included. What we do get is the background of when Euler did what, how, and why.</p>
<p>
Much about Euler's life and scientific contributions can be looked up on the Web, which I will not repeat in detail here. The way Calinger has organized the book is by subdividing the life of Euler into chapters that discuss time slices of only a few successive years. Since the focus is on his scientifically active life, the first chapter collects the first twenty years that Euler spent in Basel. Then there are four chapters about Euler's first stay (1727-1741) at the Imperial Russian Academy of Sciences in Saint Petersburg. He accepted an invitation from this Academy because he was not appointed professor in Basel as he had hoped and because that Academy attracted the best scientists from Europe, among which two Bernoulli brothers, sons of Euler's teacher Johann I Bernoulli. But when the climate became unfavorable at the Academy around 1740, Euler accepted an invitation from Frederick the Great who wanted to start a prestigious Prussian Academy in Berlin. Seven chapters deal with the apogee years of Euler in Berlin (1741-1766). Then, because the relation with the king was not optimal, he was not appointed president, even though he had done a lot for the Academy and had been a director of the Mathematical Section for many years. Therefore he accepted an invitation of Catherine the Great to return to Saint Petersburg where he stayed till the end of his life. Calinger covers the latter period with three chapters.</p>
<p>
Euler was plagued with health problems. From the well known portraits of Euler, it is obvious that he had problems with eyesight. From 1738 he lost sight in his right eye and after a cataract operation on his other eye he seized an infection so that from 1766 on he was almost completely blind. He married Katharina Gsell in 1734. During their long marriage, they had 13 children of which only five reached adulthood. When his wife died in 1773, he remarried the younger half sister of his first wife. All these facts are pretty well known.</p>
<p>
Calinger's main sources are Euler's scientific publications but even more so the surviving notebooks that Euler kept during his lifetime (some 4000 pages) and the extensive correspondence that Euler had with many of his peers. Quite often, some ideas were first formulated, or problems and solutions were discussed in letters. Only later the same ideas appeared in print in journals or were presented to the Academies or, if wrapped up in a book, that took even longer, often years, before the book was printed. As a consequence, several topics return and evolve in the successive time slices in the book. For example Euler as a sincere religious person was fighting the monad theory that has its roots with Pythagoras, but that was revived by Leibniz. So till the middle of the century Euler spent quite some energy in disputes with the Wolffians, on religious grounds. These shared their interpretation of Leibniz's <em>Monadology</em> with their main spokesman Christian Wolff. Euler argued that the monadic belief that reason is the basis of all knowledge would lead to atheism, which was unacceptable for him. This theme reappears several times in separate chapters. Euler didn't shy away from controversy and he chose sides in several involving Voltaire, d'Alembert, Clairaut, Maupertuis, König and others, and again these often span several chapters. While in Berlin, he still kept corresponding with the group in Saint Petersburg and pulled strings there as well. It happened on occasion that a lively correspondence with a person suddenly drops to zero for a longer period. Calinger then of course explains what caused this.</p>
<p>
Most chapters start with a survey of what happened in the period considered in that chapter, what publications were written or published, what problems Euler had to deal with, the general political background, etc. We also find in most chapters a section on Euler's life, with an account of his health problems, or of what happened with his family. With the detailed sources available, we learn for instance the date and even the hour of the day when Euler left Basel for his travel to Saint Petersburg. Chess and music where his only recreations when he was not working. His ability to concentrate, even in a hectic family environment is legendary. Fortunately his wife took care of all practical problems of the household, so that he could fully concentrate on his job. From his letters, we also learn that Euler has repeatedly negotiated his salary. His financial agreement with Catherine II for his second period in Saint Petersburg was outrageous with extras besides his regular salary and a survival pension for his wife. We also learn that Euler was not always the most tactical and polite opponent. Networking in the company of noblemen was necessary, but himself not being a noble (his father was a pastor), made the communication not always easy. It is explained how he organized his work when he became totally blind: he arranged for the young Nicolas Fuss from Basel to be hired in Saint Petersburg as his personal assistant, and we read that he enjoyed using his old Basel accent. We are informed about his maneuvering to acquire, much against the likings of his sons, a new wife after his spouse had died in 1773. It is made clear how strongly Frederick II personally supervised and fostered the Berlin Academy, but how he was regularly involved in wars so that his attention was diverted from the Academy. Calinger tells us how Euler came to his results. Early in his career Euler is just making a lot of computations, until a pattern starts to show. More computations are done to verify an assumption, and from this finally the abstraction and the general result could be concluded. It is also explained how Euler operated as a teacher, how he managed business at the Academies (he thought that members should be expelled when they were not productive), what prizes he won at the Paris Academy of Sciences. We learn about the wars that went on and on (where the camaraderie between officers of both sides was more important than dedication to the soldiers of their own army), even the number of inhabitants of the cities that Euler lived in are included.</p>
<p>
Of course also Euler's mathematical and other scientific achievements are mentioned. For example where and when he introduced modern notation that is still used today like the capital Σ for the summation sign, the introduction of the Euler constant e as basis of the natural logarithm, the Euler-Mascheroni constant γ, the notation <em>f(x)</em> for a function, notation for trigonometric functions such as sin, cos, sec,..., i and j were originally used for infinity large numbers, but later he used i for the square root of -1 and ∞ for infinity. The origin and discussion of famous problems connected to Euler such as the Basel problem (a 100 year old problem to sum the reciprocals of the squares of the natural numbers, solved by Euler in 1935, giving him immediate recognition), the Saint Petersburg paradox (a paradox related to probability in a lottery problem), the Bridges of Königsberg (a problem that he solved and that is considered to be the origin of graph theory), etc.</p>
<p>
The book is so rich in information that it makes it the best reference work on Euler that is currently available. There are quotations, but not too many so that the text is not a collection of quotes which sometimes happens with biographies, but it is a fluent story of a remarkable man, placed on the detailed political and scientific background of an exciting period that is populated by all these masterminds that formed modern science. The readability is obtained because short quotes are often just placed in a sentence, and not as separate quotes. The quotes and the titles of the papers and books are in the original language (with an English translation following between parenthesis). The Academy in Berlin had adopted French as the official language of science. There are many grey-scale illustrations (these are often showing the title page of a book, or the portrait of a relevant person). Reference is made to notes collected at the end of the book, where we find also the Enström index of Euler's papers, the numbering used in his <em>Omnia Opera</em>, a list of facsimile reproductions and publications of Euler's publications that became recently available, and many other sources about Euler and his work. The price one has to pay for a readable text is that it might be somewhat more difficult to find something in a dense text. To this end, the name index and general index that is provided with detailed lists of the pages where the name or the term is used is very useful. As mentioned before, for the detailed mathematical analysis like in Sandifer's <em>How Euler did it</em> columns are not to be found here (that would be a completely different encyclopedia). Nevertheless, for all other information about Euler, this book will be a standard for many years to come. </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 most complete biography of Leonhard Euler on the market so far. It summarizes and completes the already abundant material this is already available in print and electronic form. His work, his private life, the political and philosophical background, his social network, are all very well researched and documented. The book is organized in time slices partitioning his life span in successive periods, covering fewer years when many things happened and more years for the less relevant periods. The more advanced technical and mathematical details are left out so that the result is a very readable account of the life and work of a remarkable genius. </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/ronald-s-calinger" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Ronald S. Calinger</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/princeton-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">princeton university press</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">9780691119274 (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">US$ 55.00 (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">696</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="http://press.princeton.edu/titles/10531.html" title="Link to web page">http://press.princeton.edu/titles/10531.html</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/01a50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A50</a></li></ul></span>Tue, 22 Dec 2015 07:20:24 +0000Adhemar Bultheel46620 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/leonhard-euler-mathematical-genius-enlightenment#commentsHow Euler Did Even More
https://euro-math-soc.eu/review/how-euler-did-even-more
<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>
Sandifer is a renowned expert on Euler. In 2001 he was co-founder and later secretary of the Euler Society. He wrote several books and contributed and promoted the Euler centenary year in 2007. Since November 2003, he also had a monthly column called <em>How Euler Did It</em> hosted by the Euler Archive website in collaboration with the Mathematical Association of America (MAA). These columns are stand-alone items elucidating a particular aspect of Euler's work. They are available online at the <a href="http://eulerarchive.maa.org/hedi/" target="_blank">Euler Archive</a>. A first collection of these columns appeared as a book <em>How Euler Did it</em> (MAA, 2007). Unfortunately in 2009, Sandifer had to recover from a severe stroke, but some colleagues filled up some of the gap so that the column kept appearing until early 2010. This book contains 35 of these columns from March 2007 till February 2010. September 2009 does not exist for the reason just explained. October, November, and December 2009 were filled up by guest author Rob Bradley.</p>
<p>
In the previous collection, the items were ordered chronologically, but in this volume, they are grouped by topic: Geometry, Number Theory, Combinatorics, Analysis (the largest part), Applied Mathematics, and some miscellaneous part called <em>Euleriana</em>. The columns are explaining indeed how Euler did prove some of his results. Euler got his problems for example from marginal notes by Fermat who, as we know, announced theorems by scribbling some notes in the margin of a book, or some problems were formulated by Euler himself and there are many other sources. Some historical background is given, but the main contribution is just explaining how Euler indeed constructed his proof. Sometimes original drawings are reproduced. These columns are certainly welcomed by readers not familiar with the original language of the papers or the correspondence which was often Latin, German, or French. Some translations of the original texts by Euler can be found at the <a href="http://eulerarchive.maa.org/" target="_blank">Euler Archive</a> but many are not translated yet.</p>
<p>
There are too many different topics to be discussed in detail in this review. They include prime numbers, trigonometry, probability theory, mortality tables and actuarial science, the zeta and gamma functions, formulas to approximate π, partial fractions, complex analysis, optics, fluid dynamics, gravity and many more. The <em>Euleriana</em> part deals with Euler as a teacher, but there are also contributions about Euler and the hollow earth, errors that Euler made, and about Euler and pirates. I leave it to your imagination what these are all about. You will have to read the column to know.</p>
<p>
Let me take just one example from the applied mathematics chapter to illustrate how the items are treated by Sandifer in his blog. In 1756 Euler while working in Berlin publishes a paper on modeling of saws. The column starts by explaining that Euler worked in 1728 as a physician for the Russian Navy in St. Petersburg where he learned the importance on lumber and the operation of sawmills. When 30 years later the Prussian King Frederik II was about to embark in a war, Euler, remembering the importance of lumber, wrote his paper. The saw model is about a vertical blade that cuts when moving down. Each tooth should cut the same amount of wood, which means the teeth side of the saw should be slanted, the next tooth cutting another peel where the previous one had just removed its part. This models the shape of the saw. Then the motion is formalized, the middle part of the saw blade that really cuts the wood, the speed and the energy needed, and finally the manpower needed to lift the saw (it was supposed to move down by gravity), and the amount that was cut per worker and per hour. It beautifully illustrates the genius of Euler in an easily understandable mathematical language as it is brought to us by Sandifer.</p>
<p>
As you can see from my previous enumeration, there are enough topics to make the book of interest to many. Simple mathematics suffice to illustrate the brilliant mind of Euler. They are also mathematical gems as a column: well written and documented, mainly addressing mathematicians or teachers, but understandable with a minimum of training. If fits perfectly well in the MAA Spectrum series that targets the general mathematically-interested reader. Given the dynamics and volatility of websites, it is a good initiative that the MAA has made these columns available as a book.</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 monthly columns that Sandifer wrote on the work of Euler. It contains 35 of the last columns that appeared in 2007-2010 after previously a similar collection of <em>How Euler Did It</em> was published in book form by the MAA in 2007.</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/c-edward-sandifer" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">C. Edward Sandifer</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/maa-cambridge-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">MAA; Cambridge University Press</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">9780883855843 (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">£ 23.00</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">247</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="http://www.cambridge.org/be/academic/subjects/mathematics/recreational-mathematics/how-euler-did-even-more" title="Link to web page">http://www.cambridge.org/be/academic/subjects/mathematics/recreational-mathematics/how-euler-did-even-more</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/01-06" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-06</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/01a50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A50</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/05-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05-03</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/11-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">11-03</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/26-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-03</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/51-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-03</a></li><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>Mon, 20 Jul 2015 16:06:22 +0000Adhemar Bultheel46314 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/how-euler-did-even-more#commentsAndré-Louis Cholesky
https://euro-math-soc.eu/review/andr%C3%A9-louis-cholesky
<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>
André-Louis Cholesky (1875-1918) was a French military officer who served as a topographer in the army. He studied at the <em>École Polytechnique</em> and was sent to Crete, Tunisia and Algeria for measurements. He was also participating in correspondence courses at the <em>École Spéciale des Travaux Publics</em> founded by <em>Léon Eyrolles</em> in 1891. During the war he was assigned to the artillery but in the period 1916-1918 he was also director of the Geographical Service in Romania. After his return to France he took part in the second Battle of Picardy where he was fatally wounded. He died in Bagneux near Soissons at the end of WW I. Cholesky is famous among topographers, certainly in France, while most mathematicians will be familiar with the Cholesky method for the factorization of symmetric matrices. Cholesky never published the method himself, but Ernest Benoît, a military colleague, published the method in 1924. Benoît had also written a short biography before (an English translation of 1975 is included in this book).</p>
<p>
Claude Brezinski is professor emeritus of numerical analysis, but he has been keen on the history of sciences and has published papers and books in both domains. Already in 1996 he wrote about Cholesky. That was when he got access to the documents at the <em>École Polytechnique</em> that, according to the French law, became public 120 years after Cholesky's birth. His research got an enormous momentum when in 2003 he was contacted to help classify Cholesky's archives that the Cholesky family wanted to donate to the <em>École Polytechnique</em>. The result is reflected in this book.</p>
<p>
The logical start is a detailed biography of Cholesky. It has many illustrations, mostly pictures of Cholesky, and (translated) citations from several documents.<br />
Raymond Nuvet, the vice-mayor of Montguyon, Cholesky's birth place, is largely responsible for a second chapter that sketches the family history from Cholesky's great great grandfather till his grandchildren. The roots lay most probably somewhere in Poland, but the precise origin is uncertain.<br />
Chapter 3 explains some elements from topography. That involves triangulation: one has to measure all the angles of the connected triangles, but only the length of one side of a starting triangle. All unknown lengths, and hence the coordinates of the vertices, can be computed by solving a system of equations. To deal with measurement errors, a least squares solution is computed. In this context the system is usually underdetermined. Another aspect of topography is leveling. Because not all measurement points will be in the same horizontal plane, their relative elevation has to be measured and taken into account. Cholesky developed a method of double-run leveling in 1910, which is still used today.</p>
<p>
The following chapter is more extensive and deals with Cholesky's method to solve (symmetric) linear systems. The history starts with the least squares technique attributed to Gauss, the method of Gaussian elimination, variants by Doolittle and others, and of course a discussion (and a translation) of Cholesky's unpublished notes (the original French version appears as an appendix). An analysis of the notes shows the skills of Cholesky. He discussed the computational complexity, the convergence of the square root computation, and gives a rounding error analysis. Brezinski writes <em>"If this paper was submitted today to a numerical analysis journal, it would be recommended for publication without any hesitation".</em> The chapter continues by explaining how the method was re-invented, and how it gradually was spread among the computing community and what current research is dealing with (iterative methods, preconditioning, etc.).</p>
<p>
Other work of Cholesky (military and topographical) is surveyed in a short chapter, and another chapter sketches a biography of Léon Eyrolles, especially the early evolution of the <em>École Spéciale des Travaux Publics</em> where Cholesky was a professor. Eyrolles is also the founder of the publishing company <em>Éditions Eyrolles</em>. In the archives of Cholesky, also an unpublished book was found about graphical computation. It is typeset in the original French version in an appendix and it is placed in its historical context and extensively discussed in a separate chapter by Dominique Tournès who is professor of mathematics and the history of mathematics.</p>
<p>
The next chapter is devoted to Ernest Benoît. The authors had a hard time to find the correct information. In all reports he was mentioned as <em>Commandant Benoît</em> (or with another military title as appropriate). Not even an initial for his first name was known and the name Benoît is a common name in France. Nevertheless his history is tracked down and a translation of his eulogy of Cholesky is included in English translation.<br />
The last chapter is an inventory of other documents from the Cholesky archive. It contains translations of military reports about Cholesky and of diary booklets written by Cholesky during his field work in France in 1905.</p>
<p>
This book is the only book about Cholesky that is currently available. After the little bits that were available in the few publications before, the wealth of facts that is brought here about Cholesky's life and work is overwhelming and will be almost impossible to surpass. Whatever the reader did not find here can be found in the archives in the <em>Fonds Cholesky</em> at the <em>Éole Polytechnique</em> to which the authors refer if needed. The style is rather factual and there are many illustrations. All the historical and mathematical context that a reader can wish for is provided. Sometimes pushed to the extreme. For example, every name of a person occurring in the text is followed by place and date (day, month, and year!) of birth and death (if known of course). This is clearly very informative, but you can imagine that it sometimes hinders readability a bit when several names occur in the same sentence.</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>
Cholesky (1875-1918) was a military topographer and well known as such, but he is also known among mathematicians because of his method for the factorization of symmetric matrices. The book contains his biography and a family history, but it also discusses the history of his contributions to science, and a biography of other important persons connected with Cholesky. The book is the result of the inventory of the archives that became publicly available only recently. His original (in French) unpublished paper on his factorization method and a course book on graphical computation are included. Both are extensively commented and analyzed.</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/claude-brezinski" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">claude brezinski</a></li><li class="vocabulary-links field-item odd"><a href="/author/dominique-tourn%C3%A8s" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Dominique Tournès</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/birkh%C3%A4user-basel" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">birkhäuser basel</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">2014</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">978-3-319-08134-2 (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">100,69 € (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">344</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="http://www.springer.com/978-3-319-08134-2" title="Link to web page">http://www.springer.com/978-3-319-08134-2</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/01a50" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A50</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a55" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A55</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/01a60" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a60</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a90" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A90</a></li></ul></span>Mon, 20 Oct 2014 07:20:08 +0000Adhemar Bultheel45782 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/andr%C3%A9-louis-cholesky#comments