European Mathematical Society - 01A40
https://euro-math-soc.eu/msc-full/01a40
enThe secret formula
https://euro-math-soc.eu/review/secret-formula
<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 "secret formula" is a method found by Tartaglia to solve cubic equations. The 16th century priority fight between Tartaglia and Cardano's student Ferrari over this formula is well known. A brief summary: Niccolo Tartaglia, an abbaco master in Venice, was challenged in a mathematical duel by Antonio Fior, and solved the questions, all reducible to solving cubic equations, in a couple of hours. This made him instantly famous, since Luca Pacioli had stated that, contrary to the quadratic equation, there did not exist a general formula for cubic equations. Tartaglia did not want to reveal his secret formula, probably to keep up his fame and to have an advantage in future duels. Gerolamo Cardano however wanted to include the formula in his book on algebra that he was preparing. Only after a lot of nagging, Tartaglia discloses to him his method in verse form, subject to strict secrecy. When Cardano and his student Ludovico Ferrari discovered that the formula was known in unpublished work from Scipione dal Ferro, they considered the promise to Tartaglia not binding any more, and the method was included in the book with proper reference. This resulted in a public fight between Tartaglia and Ferrari publishing insulting pamphlets back and forth.</p>
<p>
Niccolo Tartaglia was born in Brescia around 1500. His last name is often believed to be Fontana, but Toscano claims that there is no solid proof for that. Niccolo preferred to use his nickname Tartaglia, i.e. the stammerer, because that is what he did since at the age of 12, a French soldier's sabre mutilated his jaw and left him for dead during a reprisal attack of the troupes of Louis XII on his home town. Against all odds, his mother could keep him alive. Later he became an abbaco teacher. That means that he instructed and applied the practical use of numerical calculation using the newly introduced Hindu-Arabic numerals as described in Fibonacci's <em>Liber Abbaci</em>, rather than the impractical Roman numerals. This practical kind of mathematics was needed in commerce for bookkeeping or for example to converse different measures or currency. It was quite different from the geometry of Euclid's <em>Elements</em> that was taught at an academic level.</p>
<p>
The Renaissance habit of having public challenges and scientific duels had some historical background. A number of questions were formulated by the challenger to be solved within a certain time span. It was a gentlemen's agreement though that no questions should be asked that the challenger was not able to solve himself. The one that was challenged then reposted with a set of questions for the challenger, and the winner of the duel was the one who first solved all the problems first or who solved most of the problems. Tartaglia received in 1530 two questions by Zuanne de Tonini da Coi, and that surprised him because the problems reduced to the solution of a cubic equation, which was claimed to be impossible by Pacioli. So he assumed that da Coi did not know how to solve it either. Nevertheless he started thinking about the problem, and obviously found a solution, at least for some cases. It should be noted that what we write in modern notation as $x^3+ax=c$ and $x^3=bx+c$ were considered to be two different types of equations. Because the terms represented (positive) quantities, (often lengths with a geometric interpretation), they could not be zero or negative. Only positive coefficients were allowed, which made it difficult to switch terms to the other side of the equal sign.</p>
<p>
To explain the state of the art of algebraic manipulation, Toscano sketches in a second chapter the history of how algebra came to Europe from the Babylonian Plimpton 322 tablet and the Egyptian Rhind papyrus to Al-Khwarizmi's Algebra book (<em>The Compendious Book on Calculation by Completion and Balancing</em>) in which is explained how to solve equations without explicitly switching terms to the other side of the equal sign. It would have been simpler if negative numbers were allowed and if our symbolic notation was used. Although the latter was once promoted by Diophantus of Alexandria (3rd century), the habit was lost over time. It is only because of the 16th century events described here that our modern notation and manipulation came about.</p>
<p>
The next chapter is describing the 1535 duel with Fior that started Tartaglia's fame. Antonio Fior challenged Tartaglia with problems that all reduced to cubic equations and Tartaglia, who had figured out how to do it since da Coi's questions, gave the answers in a few hours long before Fior could solve one of Tartaglia's problems. Whether Fior was able to solve the problems himself, is not clear since he kept begging Tartaglia to reveal his method, although he claimed that the method was explained to him by "some mathematician" 30 years ago. This was most likely dal Ferro since Fior was his assistant.</p>
<p>
But Tartaglia was now also approached by Cardano, first through his publisher, who wanted to include Tartaglia's method in his book on algebra. When Cardano invited him later to Milan, Tartaglia finally disclosed his method after Cardano had sworn not to publish it. Toscano explains the rhyme that Tartaglia used to summarize how to solve the two forms of the cubic equation mentioned above and how a third form is reduced to one of those. When Cardano's book <em>Ars Magna sive de regulis algebraicis</em> was published in 1545 it was a big success and historians consider it as the beginning of modern mathematics. It contained, besides Ferrari's solution for the fourth degree equation, also the method for the cubic equation with proper reference to Dal Ferro and Tartaglia. Tartaglia was however furious and he published his <em>Quesiti et inventioni nuove</em> containing his account of what has happened, alongside some insulting remarks about Cardano. This was published one year after Cardano's book, while he had neglected for many years to publish his method himself. Cardano had accepted a long-cherished physician's position and had left mathematics teaching, so Ferrari took up the defence of his supervisor and there was quite some verbal abuse in public pamphlets exchanged between Ferrari and Tartaglia in subsequent months. This culminated in a duel between both in 1548 in Milan, with Ferrari as victor.</p>
<p>
Cardano's book is important for the history of mathematics because it initiated some ideas that lead to complex numbers. On the other hand, the Cardano-Tartaglia-Fior-Ferrari interaction is a juicy topic that easily lends itself to be discussed in popular science books. So it has been told by many authors, but it is often Cardano who is placed at the center of the story. For example in P.J. Nahin <a href="/review/imaginary-tale-story-%E2%88%9A-1" target="_blank"> <em>An Imaginary Tale: The Story of √-1</em></a> (1998/2010) some time is devoted to this melee and in the novel by M. Brooks <a href="/review/quantum-astrologers-handbook" target="_blank"> <em>The Quantum Astrologer's Handbook</em></a> (2017) Cardano is the main historical character. In the present book, it is basically the same story all over, but told more from Tartaglia's viewpoint. A lot is taken from Tartaglia's own account with many translated quotations in which the mutual scolding in the pamphlets are made blatantly clear. There is of course some background and history of mathematics but Toscano's main focus is the solution of the cubic equation leaving some other work of Tartaglia and Cardano in the shadow. For example Tartaglia wrote a treatise on ballistics and found that the maximum reach was obtained firing in an angle of 45 degrees. The result is correct although it has several mistakes for which Ferrari reproached him later. He also translated Euclid's <em>Elements</em> to Italian (working on this was his excuse for not publishing his formula).</p>
<p>
Toscano has a pleasant writing style (and/or the translation by Arturo Sangalli is smooth). The opener of the books describes Niccolo with his mother and sister lost in the chaos of French soldiers attacking Brescia. Niccolo is hit twice by the sabre of a soldier and left for dead. That is like the opener of a dramatic novel. The attention of the reader is immediately caught. As Toscano unravels the historical development, he makes use of many quotations, which are fortunately provided by Tartaglia himself. This implies that the story is told close to how Tartaglia has experienced what has happened. However Toscano does not hesitate to give some interpretations and place some question marks where appropriate. The yeast of the story has been told already many times, but it has never been told like Toscano does in this 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 translation of the Italian original published in 2009. On the background of 16th century Italy, Toscano describes how Tartaglia has learned how to solve cubic equations, thus winning in a spectacular way a mathematical duel against Antonio Fior. Tartaglia does not want to share his method with others, but eventually he lifts a tip of the veil for Cardano subject to strict secrecy. Cardano publishes it anyway because he discovered that the formula was described in older unpublished work of Dal Ferro. This results in a fierce public pamphlet war between Tartaglia and Cardano's apprentice Ferrari.</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/fabio-toscano" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Fabio Toscano</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/princetion-university-press" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Princetion 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">2020</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">9780691183671 (hbk), 9780691200323 (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">USD 24.95 (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">184</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://press.princeton.edu/books/hardcover/9780691183671/the-secret-formula" title="Link to web page">https://press.princeton.edu/books/hardcover/9780691183671/the-secret-formula</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/01a70" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01a70</a></li></ul></span>Mon, 15 Jun 2020 09:29:42 +0000Adhemar Bultheel50829 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/secret-formula#commentsThomas Harriot
https://euro-math-soc.eu/review/thomas-harriot
<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>
As a young mathematician and astronomer, Thomas Harriot (c. 1560-1621), was hired by Sir Walter Ralegh, to train the captains of the ships ready to cross the Atlantic and claim some territory for England in the New World. Ralegh was a poet, politician, and frequenter of the court of Queen Elisabeth I. The faith of the two man was entangled since. A first expedition brought two native Americans back to England and Harriot learned their language and invented a phonetic alphabet to write it down. He even crossed the ocean himself to visit their home land during a second expedition. After his return, he wrote the only book published during his life: <em>The Briefe and True Report of the New Found Land of Virginia</em> (Virginia named after Elisabeth, the virgin queen). He continued working for Ralegh as a land surveyor. Later Harriot got employed by Henry Percy, 9th Earl of Northumberland. With observations and tedious computations he improved his skills as an astronomer, a mathematician, and a physicist.</p>
<p>
Elisabeth, the short tempered virgin queen, depended on trusted advisors like Ralegh and Percy, but when Ralegh got married without her consent, she considered it treason. The period was turbulent with many political en religious tensions for example with the catholic Queen Mary of Scots, and hence also with France and Spain resulting in naval battles. Privateering was a lucrative pastime for the crew of the ships that were exploring the New World. Political compromises and treason, spying and conspiracies, were common games for Queen Elisabeth and her successor King James. Take on top of that the emergence of the scientific revolution of the 17th century, and the exploration of the North and South Americas, and Arianrhod has all the ingredients to write a thrilling and adventurous novel about it, and so he did. Only it is all based on true and well documented facts.</p>
<p>
Arianrhod has found a good balance between explaining the mathematical and astronomical work of Harriot, and sketching what happened on a political and personal level of the main characters that he describes in the turmoil of events at the end of the 16th and the early 17th century. These main characters whose fortunes and misfortunes are told, are besides Harriot mainly Ralegh and to some extend also Percy but there are many others as well. Too many to keep track of if this were a fiction novel, but real life is not that simple. A name list in the appendix with one line description per name might have been welcome for a reader not so familiar with the period, its politics and its science.</p>
<p>
It is characteristic of many biographers that they bring an idolatrous glorification of their subject. Somehow this struck me in this book too. It is clear that Arianrhod describes Harriot and Ralegh as the worthy heroes with almost sacred virtues. For example the attitude that Harriot and Ralegh have towards the native Americans, considering them as friends and treating them with respect, leaving their dignity is unusual for that time. The devotion of Ralegh for his queen, even when she locked him up in the Tower of London for many years (as she also did with Percy) is outspoken. His continued attempts to colonise parts of North and South America to flatter the Queen (which both turned out to be disastrous) and the noble way he behaved on the scaffold when he was beheaded under King James I are almost beyond human limits.</p>
<p>
Harriot fell under the bad faith of his employers when they were accused of treason, and that shone on him and his work and ideas were scrutinized for possible atheistic elements. He was even imprisoned for a short while on the charge that he had cast an horoscope on King James during the Gunpowder Plot. However, he always tried to stay somewhat in the shadow, concentrating on his scientific work, and so he escaped most of the misfortunes that befell on his employers and could have been his faith too. This is one element of excitement, but the whole book is a thriller: will the pioneers survive crossing the ocean, will they survive in the midst of unknown tribes, will the prisoners of the Tower be executed, will the catholic Gunpowder Plot or the courtier's Main Plot against King James be a success, will the war at sea with Spain be won, will the money be raised for yet a new expedition,... all components that can bring some tension in an engaging story told with brio. That is why, besides Harriot, Ralegh, and Percy, many other (mostly political and scientific) characters are staged in this complicated interplay of intrigues. And Arianrhod is not the first to use these elements in a book. He suggests that Shakespeare has used some of the events described here as scenes in his plays.</p>
<p>
When it comes to Harriot himself, these parts of the book are mainly about his scientific achievements. The reader is instructed about how Harriot explained the seamen what they should know to find the position of his ship, and how they could compute it in an efficient way. We learn about his study of the loxodromic curve and the equiangular spiral, how he unravelled the secret of the Mercator projection, how he measured the acceleration of free falling objects to study gravitation and how he computed the trajectory of a canon ball (that was before Newton formulated his laws of motion). Harriot studied the precession of the Earth and the Gregorian calendar, the refraction of light and gave an explanation for the rainbow. We read about his atomistic views, his exploration of probability theory, and of course his astronomical computations and his study of sunspots. He produced a map of the moon and he had some correspondence with Kepler. He was also one of the first to simplify algebra by introducing symbolic notation: he used letters for variables, he had a notation for exponents, a variant of our equal sign,... Arianrhod places all his discoveries in context, sometimes going back to Greek antiquity or by discussing contemporaries or scientists who came after Harriot who discovered the same things independently, and whose name became attached to these results.</p>
<p>
It is a shame that Harriot did not publish more because that would certainly have given him a reputation comparable to Galileo, Kepler, and cartographers and mathematicians of his time. When at the age of sixty he seemed to be ready to start rounding up his work and publish it, he got health problems. It might have been a kind of cancer that started with his nose that finally killed him. It is not unlikely that it was caused by excessively smoking tobacco. Allegedly he, and his employer Ralegh, are responsible for introducing pipe smoking in England. He made his testament and asked some friend to order his notes and publish them posthumously. However not much came out of that. His notes got lost until they were rediscovered at the end of the 18th century, but again, it took a long time before it was recognized that Harriot had a lot of results that we now know by the name of other scientists, while Harriot had these already much earlier. Slowly Harriot's achievements were realized by historians analysing his notes that are now fully digitized and made available through the <em>European Cultural Heritage Online</em>.</p>
<p>
To conclude: this is a marvelous book because of the engaging way it is told, very much unlike a dull biography with an enumeration of facts. Moreover it is also well documented by additional material to be found in the last 100 pages of the book. There you can find a number of graphic illustrations that are needed to understand some of the mathematics that are discussed. These are moved to the appendix probably to allow the reader to skip some of the mathematics if he or she is not interested. Many of these extra pages are filled with notes that explain some background or give the origin of a quote or a justification for a statement in the text. Of course the list of primary and secondary sources used are there too, and a well stuffed index of names and subjects. A warmly recommended read about England in the period of Shakespeare, shortly before Newton, with in the background a turbulent dance of politics, when war could still be avoided by marriage, but fighting over colonizing the Americas, and over religious controversies never ceased. On this canvas Arianrhod paints the bubbling emergence of the Scientific Revolution to which Harriot was a silent contributor.</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 biography about the life and work of Thomas Harriot (c.1560-1621), an English astronomer and mathematician. Because he did not publish much, most of his work has been hidden for long time but since the legacy of all his notes was rediscovered, historians of the 19th century and later have found that his work preceded in several ways results by mathematicians like Galileo, Kepler, and even Newton. This book makes these insights available for a general public.</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/robyn-arianrhod" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Robyn Arianrhod</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">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">9780190271855 (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">£ 19.99 (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">376</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://global.oup.com/academic/product/thomas-harriot-9780190271855?cc=be&amp;amp;lang=en&amp;amp;" title="Link to web page">https://global.oup.com/academic/product/thomas-harriot-9780190271855?cc=be&lang=en&</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/01-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-02</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/01a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A40</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>Mon, 14 Oct 2019 09:05:19 +0000Adhemar Bultheel49811 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/thomas-harriot#commentsThe Quantum Astrologer's Handbook
https://euro-math-soc.eu/review/quantum-astrologers-handbook
<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>
Gerolamo Cardano (1501-1576) was an Italian polymath. He studied medicine and earned reputation and wealth as a physician, but he was also gifted mathematically and he used negative numbers and imaginary numbers as square roots of negative numbers well before they were more generally accepted. As a gambler, he also laid foundations of probability theory a century before Fermat and Pascal worked it out and 200 years before Laplace finished the job.</p>
<p>
Michael Brooks has a PhD in quantum physics from the University of Sussex, but he switched career and became a journalist and most of all a science writer. In this book (his seventh) we learn how his science is entangled with what Cardano (or Jerome as Brooks calls him) did. It has been remarked before by the authors of the papers collected in <a href="/review/art-science" target="_blank"> <em>The Art of Science. From Perspective Drawing to Quantum Randomness</em> </a> (Springer 2014) that Cardano's findings (complex numbers and probability) laid the foundations for the elements that are so essential for the development of quantum theory. Brooks is exploring this trace by writing a novel-like biography of Cardano, and at the same time explaining his own field by discussing the history and the subtleties of quantum physics and the many questions that it has raised and that still remain unsolved even today.</p>
<p>
Collecting the data for a biography of Cardano is not a major problem since he wrote an autobiography near the end of his life. Born as an illegitimate son of Frazio Cardano, a jurist and mathematician, his birth is almost a miracle because his mother tried abortion, but he got born anyway and survived frequent illness and the plague to which his three siblings succumbed. He decided to study medicine against his father's desire. He applied several times to be accepted as a physician in Milan, but it was repeatedly refused which caused him to live in poverty. By the mediation of some influential friends he got eventually a professorship in mathematics in Milan and got his medical licence. He was now a respected scientist and the most popular physician of Milan. He got offers from kings of Denmark, France, and the Queen of Scotland, that he all refused. He did travel to Scotland though where he also treated the archbishop John Hamilton, whom would later save Cardano from the inquisition. With his wife, Lucia Brandini, who was the love of his life, he had three children but his oldest son got executed for poisoning his own wife, and he disinherited his youngest who was a gambler stealing from his father. His outspoken confrontational ideas (his book <em>On the Bad Practice of Medicine in Common Use</em> was a success but not gracefully accepted by colleagues) and influential jealousy of his success brought his reputation down and allegations about his behaviour made the inquisition decide to imprison him at the age of 69 for the obscure reason of having cast an horoscope of Jesus Christ. So he had to spend several months in jail. By an intervention of John Hamilton he got out but he lost his professorship and was forbidden to publish his work. Not feeling accepted in Milan anymore, he moved to Rome where he wrote his biography. He predicted his own death on September 21, 1576 presumably by committing suicide.</p>
<p>
Brooks has interwoven this biography with the evolution of quantum physics by using a fictional component in which he is visiting Cardano while he is imprisoned waiting for his release or conviction. Cardano writes in his biography that he was visited by a guardian angel, and Brooks is taking up this role and they have a conversation of which this book is a reflection. We learn about the ups and downs in Cardano's life, the love of his life, the misery he has with his children, and the well known dispute with Tartagli and del Ferro about revealing the formula to solve a cubic equation. At the same time Brooks explains to Cardano (and thus also to the reader) the principles of quantum physics. He writes:</p>
<blockquote><p>
Jerome's views on astrology mirror our own on quantum physics. In quantum experiments we see things appear in two different places at once, or an instantaneous influence over something that is half a world away. We cannot make sense of it, but we don't dismiss it as ridiculous. We have the evidence of our experiments, after all, just as the astrologers have the 'evidence' of experience. (p.22-23).</p></blockquote>
<p>
Quantum physics is real as Brooks describes his history and evolution, but we still do not understand why the experiments give the results they do. He goes through all the possible interpretations from the Copenhagen interpretation to the multiverse theory and the superdeterministic interpretation, the pilot wave theory, the Penrose interpretation, etc. Cardano (and the reader) learns all about the main protagonists, the double slit experiment, Schrödinger's equation, the EPR thought experiment and its verification, and even some particle physics and string theory.</p>
<p>
</p>
<p>
Because in his conversation with Cardano, Brooks, coming from the future, knows things that did not happen yet. However, using the mysterious possibilities that quantum physics provides, Brooks can convince the reader to accept these anomalies. So the following twist comes as a surprise, and I think it is an amusing find. Brooks suggests to Cardano in prison that all this misfortune is the result of Tartaglia's doing. But Cardano answers that he doubts that because Tartaglia is dead for more than a decade. Then Brooks realizes that he read that in a book by Alan Wykes <em>Doctor Cardano</em> (1969) but Wykes may have used this historical flaw for the sake of his story. So it leaves Brooks blushing with shame in front of Cardano. At this moment Brooks is simultaneously a character in his own book and the biographer of Cardano who is correcting another biographer about historical facts. Later a similar trick is used when Brooks suggests that Cardano should write to Hamilton for help. It is then Cardano who doubts that Hamilton is still alive. But Brooks insists since he knows who has helped Cardano to get out of prison.</p>
<p>
The parallels between Cardano and Brooks, and the similarities between Cardano's science, inventions, and philosophy and the modern quest to explain quantum physics is very inspiring. Many of the findings that Cardano pioneered were way ahead of his time. For example his idea of the <em>aevium</em> even hints at a higher dimensional universe. Whatever the eventual faith or the proper interpretation of quantum physics may be, currently we are still in the dark. Perhaps we shall look in a century upon our present guesses and beliefs like we now look upon Cardano's astrology and his horoscopes, that were fully rational to him, as much as they are unscientific to us. So it may be symbolic when near the end of the book, Cardano steps out of his prison cell, leaving Brooks behind sitting on the bed.</p>
<p>
As far as I know, there is no biography-novel-popular-science-or whatever-you-call-it book produced that mixes all these ingredients in a marvelous plot. There are of course very good historical novels that sketch a biography of some scientist or another historical figure (and some exist some for Cardano already), but none has mixed this with popularizing science in such an harmonic and entertaining way as Brooks has achieved here. A novel, a biography and a popular science book, none of these in a strict classical sense, and yet all of them at the same time. Its format is certainly original. A recommended read.</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 biography of Cardano and a popular science book on quantum physics, brought in the form and with the quality of a good witty novel. Brooks takes on the double role of a character in his book, playing the role of Cardano's guardian angel who has a dialogue with Cardano while he was imprisoned by the inquisition having cast an horoscope of Christ, and at the same time he is telling the reader the ups and downs of Cardano's life and explaining his own work as a quantum physicist to both Cardano and the reader, showing some remarkable parallels between ideas of Cardano and quantum theory.</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/michael-brooks" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Michael Brooks</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/scribe-uk" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Scribe UK</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">978-1-911-34440-7 (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">£ 16.99 (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">256</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/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://scribepublications.co.uk/books-authors/books/the-quantum-astrologers-handbook" title="Link to web page">https://scribepublications.co.uk/books-authors/books/the-quantum-astrologers-handbook</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/01a09" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A09</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/00a15" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00A15</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A40</a></li><li class="vocabulary-links field-item even"><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 odd"><a href="/msc-full/81-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">81-03</a></li></ul></span>Tue, 07 Aug 2018 12:01:28 +0000Adhemar Bultheel48636 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/quantum-astrologers-handbook#commentsAll sides to an oval
https://euro-math-soc.eu/review/all-sides-oval
<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>
It is not difficult to define mathematically what an ellipse is. Its Cartesian equation is well known. It is however less clear what an oval is. Most people will come up with the condition that it looks like an ellipse. It is a smooth convex closed curve of the plane with two orthogonal symmetry axes. But how to be more precise? Since antiquity, ovals have been used in architecture. So what was the construction used by the architects?</p>
<p>
There are the Cartesian and the Cassini ovals, that have a simple Cartesian equation, but they do not always have the symmetry (in th first case) or are not convex (second case). Historically however the ones that have been used most in arts, especially in architecture, are the polycentric ovals that consist of circular arcs that are stitched together in a smooth way. This is the only kind of ovals that is considered in this book.</p>
<p>
The easiest and most popular one that has been studied thoroughly is the so-called four center oval. It consists of four circular arcs that fit together smoothly. Two arcs with smaller radius are at the tops of the long symmetry axis and the ones at the end points of the shorter symmetry axis have lesser curvature because they belong to a circle with a longer radius. The crux is of course to choose these four centers of the circles in such a way that the arcs fit smoothly together at the four connection points. How does one have to select these centers and how large should the sector angle be that supports the arcs so that one does indeed get this smooth transition? Because of the symmetry, two centers are located symmetrically on the long axis and two on the shorter one. So it suffices to consider only a quarter of the oval and find two of the centers to define the arcs and the connection point where the arcs meet in that quadrant. Once the length of the axes are given, an ellipse is completely defined. For an oval, one needs at least one more parameter, like the distance from one of the centers of the circles to the center of the oval or the distance of the connection point to one of the symmetry axes.</p>
<p>
Once these arguments have been formulated, it needs some analysis of the geometry of the problem. And that is where this book gets started. The author has, besides other interests, a knack for polycentric curves like eggs or ovals. This book is restricted to ovals, and the first chapter analyses the properties that will enable us to relate the different parameters. Once this is cleared out, the construction with ruler and compass of an oval (actually a quarter of an oval, because the rest follows by symmetry) is given step by step depending on which parameters are prescribed. So one might choose three of the six possible parameters in many different combinations and that gives rise to twenty different ways to define and construct an oval satisfying the data. Some are more complex and some have more restrictive conditions than others. The solution may not always be unique. Everything is clearly explained and the many illustrations produced with geogebra are crystal clear. It might however be interesting to have a look at the associated website <a href="http://www.mazzottiangelo.eu/en/pcc.asp" target="_blank">www.mazzottiangelo.eu/en/pcc.asp</a> where you find links to YouTube videos showing animated geogebra constructions. The link goes both ways: you may consider this book as a manual for the online site, or the online site as an illustration for the book.</p>
<p>
Besides the parameters described above, one might also choose for one of the radii of the arcs or the ratio of the axes or the angle formed by a symmetry axis and the line joining the circular centers of the arcs. With all ten parameters, there are a total of 116 possibilities to construct the ovals, many of which, but not all, reduce to the twenty constructions mentioned before. Some of the constructions are historical and often pretty old, but others are surprisingly recent. For particular choices of the parameters, the construction may simplify considerably or the oval may have especially pleasing esthetic properties, which are discussed in a separate chapter.</p>
<p>
Towards more practical applications of stadium design, one may consider ovals circumscribing or inscribed in a rectangle. If the symmetry axes are the middle-lines of a rectangle and the diagonals of a rhombus, then all previous constructions circumscribe the rhombus and are inscribed in the rectangle. For a stadium one should find an oval circumscribing the inner rectangular field (for example a soccer field) and surround it by ovals like running tracks, all inside an outer rectangle defining the limitations of the stadium. Modern constructs however have straight parts for the running tracks along the long sides.</p>
<p>
While the constructions are mostly obvious, it takes more algebra and more formulas to express some parameters as a function of others. This is a short chapter, but essential to find ovals that are optimal in some sense. For example finding the "roundest" oval with given axes. They are also needed in geogebra animations when slider rules are provided allowing to see the effect of changing a parameter.</p>
<p>
The last two chapters discuss ovals in two famous architectures in Rome: the dome of the church <em>San Carlo alle Quattro Fontane</em> by the architect Borromini and the ground plan of the <em>Colosseum</em>. A careful study is made of the ovals of the base of the dome in the church, the rings of coffers, and of the lantern. It turns out that there are small defects making them deviate from perfect mathematical ovals. This has long been a mystery. It is suggested that the starting point was a mathematically perfect oval, but that practical restrictions entailed heuristic corrections. The solution that Mazzotti proposes here corresponds remarkably well with Borromini's original drawings.<br />
For the Colosseum, we have to leave the simple ovals with four centers and go to quarter ovals consisting of more than two arcs. Because of symmetry there have to be always $4n$ centers. Again constructions of such ovals are considered. In the case of the Colosseum, $n=2$, i.e., ovals consisting of eight circular arcs seem to match the ground plan perfectly well.</p>
<p>
This is a very nice geometric application that requires only simple algebra and that can be easily experimented with. You do not need to be a mathematician to enjoy it. It that sense, it might be interesting to have the geogebra source available somewhere, which is unfortunately not the case. Also historians might be interested in the last two chapters about historical buildings. For the mathematician, it is invaluable because it brings together so much information that was either not known or never writen down or if it was, then at least it was scattered in diverse publications. The graphics are very readable since they use colors (except for the pictures in the last two chapters, only red, green, and blue suffice for the mathematical constructions). As a LaTeX purist, I cannot resist mentioning my irritatin when seeing variables mentioned in roman font when in a sentence, while they are in a different font when used in a formula. Also, I do not understand why the ratio of the half symmetry axes is denoted at least twice as $\frac{p=\overline{OB}}{\overline{OA}}$ (p.20 and 148) and when at the end of a line $p=\frac{\sqrt{2}}{2}$ is split into $p$, which is left dangling at the end, and $=\frac{\sqrt{2}}{2}$ at the beginning of the next line (p.102). These are however minor flaws in an otherwise nice text, and as I am sure, these will disappear in a next edition. Do not let this prevent you from reading this most enjoyable book and you should certainly try out some of the constructions for yourself, either with ruler and compass or with geogebra.</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 book restricts to polycentric ovals, which means that they concist of circular arcs that fit smoothly together. Some properties are derived to allow for many different ruler and compass constructions. The major part of the book is about the case of simple ovals, i.e., ovals concisting of four arcs. They can be constructed when 3 parameters are given (like location of the four centers, the length of the symmetry axes or the location of the points where the arcs meet). The book ends with the discussion of ovals in two historic buildings in Rome: the dome of the <em>San Carlo alle Quattro Fontane</em> church by Borromini and the ground plan of the Colosseum.</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/angelo-alessandro-mazzotti" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Angelo Alessandro Mazzotti</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/springer-international-publishing" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer International Publishing</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">978-3-319-39374-2 (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">31,79 € </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">170</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/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/gp/book/9783319393742" title="Link to web page">http://www.springer.com/gp/book/9783319393742</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/51-geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51 Geometry</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/51-01" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-01</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/51m04" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51M04</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/00a65" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a65</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><li class="vocabulary-links field-item odd"><a href="/msc-full/01a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A40</a></li></ul></span>Wed, 22 Mar 2017 08:49:13 +0000Adhemar Bultheel47565 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/all-sides-oval#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#commentsThe Invention of Science. A New History of the Scientific Revolution
https://euro-math-soc.eu/review/invention-science-new-history-scientific-revolution
<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>
Modern science is born from a scientific revolution that took place after the dark Middle Ages and during the Renaissance. Wootton pinpoints the start and the end of that period by attaching two particular events to them. The start was when Tycho Brahe observed a supernova (1572) and the end coincides with the publication of Newton's <em>Opticks</em> (1704) in which Newton showed how white light could be split up in composing colors. Of course these two events are symbolic since the transition has been much more fuzzy, but compared to the non-evolution since the ancient Greek, this is rightfully considered a revolution.</p>
<p>
A main point that Wootton wants to make is that when studying the history of science, it is not easy to avoid the pitfall of looking at it in retrospect with our modern ideas and concepts. The true revolution lies in the fact that things were done differently from what had been inherited from the ancient Greek. Science did not advance from pure abstract reasoning anymore. People started exploring beyond the boundaries of what was already known. Things were so different that they were done for the first time, and that gave rise to new ideas and concepts. Concepts for which no proper words existed before, and other existing words got different meaning. Even the word 'science' with the meaning of a substantial theory based on a large amount of evidences, could previously only be applied to astronomy. Only astronomy was understood sufficiently well to make reasonably accurate predictions, even though the Ptolemaic model of our solar system was wrong. To make clear that new concepts and the words coining them were invented, Wootton in several of the chapters gathers the scientific achievements and the ideas of the scientists of this period around such an emerging concept. For example the introduction ponders on the terms "scientific" and "revolution" and how we came about to refer to this period as the "Scientific Revolution".</p>
<p>
To give some more examples, let us quickly scan the contents. Columbus discovering America (1492) and Copernicus proposing a heliocentric solar system (1543) were the start, but the true acceptance that antipodes did exist and that the Copernican system was real, came only in the seventeenth century. Wootton attaches to this a discussion of whether the laws of nature and mathematics are discovered or invented.</p>
<p>
A second part focusses on "seeing". It starts with the invention of the perspective in graphical art, which is in fact also related to measuring the height of a distant tower, but also in astronomy when studying distances between the planets. The construction of the first telescopes and finding the actual location of celestial bodies, observing the mountains and crates on the moon, and the discovy of the moons of Jupiter, destroyed the Ptolemaic model and the idea of an ideal spherical moon and stars residing on crystal spheres. For the astronomical results, the historical lead role is played by Galileo. But one also started looking at small scale too by using the microscope. In Galileo's time, the main occupation of Italian mathematicians was bookkeeping. That was about the only mathematical application in real life. Mathematics itself mainly existed as an abstraction and was considered complete and finished. However scientists started studying the physical reality by placing it in an abstract setting. The geometry involved in perspective drawing for example made mathematics and art walk hand in hand. The golden section became an ideal number, Kepler used polyhedra for his first celestial model, and Da Vinci draws the Vitrivius man with ideal ratios, double-entry bookkeeping was introduced and ballistics also involved mathematics. The world was being mathematized, and this was of course an essential element in the development of the revolution in the making.</p>
<p>
The third part is essentially devoted to linguistic issues. In fact our vocabulary when we talk about science is mainly coined in the seventeenth century. Wootton analyses in detail how our current meaning of words such as 'fact', 'experiment', 'law', 'hypothesis', 'theory', 'evidence', 'judgement' came about or how it evolved in that period.<br />
Facts were no longer ignored when they did not match the theory. Brahe collected the most accurate set of astronomical observations. Wootton attributes the coining of the English word 'fact' not to Francis Bacon as it is often done, but to his secretary Thomas Hobbes.<br />
The weight of the air was measured in one of the first carefully designed 'experiments' by reading the height of mercury in a glass tube by Pascal, although it was described earlier by Descartes. This brings along the work of Torricelli, Boyle, the experiment with the Magdeburg hemispheres, etc. Experiments also showed that alchemy didn't work and never resulted in the transformation hoped for. Witchcraft was abandoned too.<br />
The concept of 'law' as in a 'law of nature' has a longer history, and Descartes was certainly not the first to adhere to the laws of nature. Nevertheless there is a long list of laws discovered during the Scientific Revolution: Stevin's law of hydrostatics, Galileo's law of falling objects, Kepler's laws of planetary motion, Snell's law of refraction, Boyle's law of gasses, Hooke's law of elasticity, Huygens' law of the pendulum, Torricelli's law of flows, Pascal's law of fluid dynamics, Newton's laws of motion, all sounding very familiar to us.<br />
A 'hypothesis' is related to experiments in the sense that it has to be confirmed by ample observations before it can become a 'theory'. In this sense it was Descartes who established this meaning. The indisputable certainties of the old philosophy was replaced by facts and evidence from experiments that generated reliable hypotheses and even incontrovertible theories.<br />
Concerning 'evidence' and 'judgement' there is the relation with these concepts in jurisdiction. Concerning the latter, there was a difference between the English system (common law relying on a jury system) and the continental, say French, system (Roman law system, based on rigor). Some historians claimed that this is reflected in an experimental approach of the English versus the deductive mathematical approach of the French, but Wootton does not aggree. There were scientists of both types at both sides of the Channel. Nevertheless, it cannot be denied that there is still today some difference in style.</p>
<p>
Part four reflects on the immediate consequences of the Scientific Revolution. After a discussion of what a machine actually is, Wootton explains that based on the old atomistic view Descartes and others saw nature as a collection of separate objects whose interactions were controlled by mechanical laws, like a machinery. Mechanical tools and clocks existed much earlier, but the idea that this mechanical view of nature could be applied to biological systems such as animals was new. If humans were considered a special kind of animal, this leads to atheism. Of course these mechanical laws could only be applied to the material world, but also the belief in witchcraft, demons, poltergeists, fairies and the likes were removed under influence of the new science.<br />
This brings Wootton to his main point: the Scientific Revolution is the most important event since the Neolithic Revolution. The undeniable important Industrial Revolution was a consequence of the scientific one. It has been a dispute among historians whether science has contributed to the invention of the steam machine, and hence to the industrial advancement it caused. It is Wootton's conviction that it really did. Papin did indeed work with Boyle and Huygens and was a professor of mathematics. He did not succeed in producing a working steam engine, but Newcomen succeeded and only because science came first and technology was a consequence.</p>
<p>
In a concluding part, Wootton goes in discussion with other historians and philosophers of science. He gives for example arguments against the wrong arguments for a relativist account of science. He does not choose sides. He writes "This book will look, I trust, realist to relativists and relativist to realists: this is how it is meant to look". He also inspects the claim that any history of the Scientific Revolution must be Whig history. His argument is that the opponents just define history in such a way that change cannot be discussed.</p>
<p>
The subtitle of the book is <em>A New History of the Scientific Revolution</em> and the "New" is indeed justified. It is certainly not a repetition of the facts that can be found elsewhere. It gives a new and personal vision on this period with philosophical foundations and a philological analysis. The achievements and the stories of the scientists as well as the scientists themselves are abundantly represented. And that are not only the scientists of the period considered, but also historians and philosophers who have written more recently about the matters discussed. The book is extremely well researched and documented. There are many illustrations, and besides the many footnotes there are 80 pages of more extensive notes at the end. The bibliography has 67 pages and the index alone takes 48 pages. This is bound to become a basic reference in the future.</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 thoroughly researched and documented book about the importance of the Scientific Revolution that took place at the end of the sixteenth and during the seventeenth century. It is not only an historical account but at the same time a philological and philosophical analysis of this period.</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-wootton" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">David Wootton</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/allen-lane-penguin-books" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Allen lane / Penguin 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">9781846142109 (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">784</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.penguin.co.uk/books/the-invention-of-science/9781846142109/" title="Link to web page">http://www.penguin.co.uk/books/the-invention-of-science/9781846142109/</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/01a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A40</a></li></ul></span>Fri, 27 Nov 2015 16:58:50 +0000Adhemar Bultheel46571 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/invention-science-new-history-scientific-revolution#commentsJohn Napier: Life, Logarithms, and Legacy
https://euro-math-soc.eu/review/john-napier-life-logarithms-and-legacy
<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 Napier (1550-1617) is a Scottish scientist who is probably best known for his invention of the logarithms. He also made a meticulous analysis of the Apocalypse and he propagated the use of salt as a fertilizer in farming. In his time he had the reputation of an alchemist and a magician having bonds with the devil. But that was a common trait attributed to scientists in those days.</p>
<p>
Born in a well respected family, he became the 8th Laird of Merchiston. Being a devote Protestant, he found in the The Book of Revelation the proof that Catholicism was the quintessence of evil and that the Pope was the Antichrist. This must also be seen in the context of the Spanish Armada's attempt to invade England in 1588 and Spain was the Catholic enemy `par excellence'.</p>
<p>
In a first chapter, Havil gives a survey of Napier's life, to switch in the second chapter to a discussion <em>A plain discovery of the whole revelation of St. John</em> (1593), the first book by Napier. The apocalyptic last book of the New Testament is a collection of psychedelic visions that allow for several different interpretations. Napier believed it was a true account of historical facts and that it contained prophesies, i.e., predictions of the near future. Napier's book consists of propositions in which symbols of Revelation are linked to historical facts and persons. Proofs and demonstrations are given to make these statements acceptable. The seven seals border seven year periods spanning the period from Jesus Christ (29 CE) till the destruction of Jerusalem in 71 CE and the seven trumpets border subsequent spans of 245 years till 1541 CE with the start of the Reformation. He calculated the end of the world (like many others did with varying results) to happen in 1688 or 1700. Because his book was written in English (and not in Latin) it had great impact and certainly after its translation in French, Dutch, and German, Napier's reputation was established, not only in the British Isles, but on the Continent as well.</p>
<p>
Chapter three is about Napier's second book: <em>Mirifici Logarithmorum Canonis Descriptio</em> (1614). This contains the first logarithm tables with their definitions and an explanation on how to use them. These tables consist of "artificial numbers" which Napier called logarithms (literally "ratio numbers") they are used to transform multiplication into addition and division into subtraction. Originally the ratios referred to the ratios of lengths of triangles appearing in the definition of trigonometric functions in a goniometric circle, but Napier later realized that this was an unnecessary restriction. Napier chose the radius of the circle to be 107 and the logarithm of that number was 0. If we denote Napier's logarithm as NapLog(<em>x</em>) then this corresponds to what we would now recognize as 107ln(107/<em>x</em>)=107log1/<em>e</em>(<em>x</em>/107). Thus not exactly in basis e but in basis 1/e. Havil explains how Napier came to this concept and how the tables can be used. The importance of this computational tool was quickly realized and used widely. Henry Briggs, lecturing in London, adopted the tables with enthusiasm and became a good friend. He also solved the annoying problem that multiplying with a power of 10 did not show easily in the NapLog by shifting the digits or by adding a power of 10. Of course, that is the germ of what we now recognize as the Briggsian logarithms that work in base 10. Note that Napier never considered the log function. The tables were just tools for goniometric and other computations.</p>
<p>
It was Napier's second son Robert (from his second marriage) who published the <em>Mirifici Logarithmorum Canonis Constructio</em> (1619) two years after his father's death in April 1617. This is the subject of chapter four. Havil does here an excellent job in explaining in a way that it is understandable for our 20th century knowledge how the tables were constructed and how they are linked. In a later chapter about Napier's legacy, he also shows how Napier touches unwittingly upon calculus and the relation with the natural logarithm mentioned above is also revealed there.</p>
<p>
There are other computational inventions by Napier that were important in his time, but whose importance, unlike the logarithms, has faded today: the <em>Napier rods</em> also known as <em>Napier bones</em> (they were made of ivory) and the <em>Promptuary</em>. These tools, and how to use them was explained in Napier's last publication <em>Rabdologia seu numerationes per virgulas libri duo</em> (1917). Napier was inspired by the <em>gelosia</em>. That is a tool that helps multiplying two integers. For example to multiply 72 x 35 = 2520, one constructed.</p>
<p>
$$<br />
\begin{array}{c|c|c|c}<br />
&{\color{red}7}&{\color{red}2}&\\\hline<br />
& 2/\ & 0/\ & \\<br />
{\color{blue}2} & \ /1 & \ /6&{\color{red}3}\\\hline<br />
& 3/\ & 1/\ & \\<br />
{\color{blue}5} & \ /5 & \ /0&{\color{red}5}\\\hline<br />
& {\color{blue}2} & {\color{blue}0} &<br />
\end{array}<br />
$$</p>
<p>
The 72 goes on top, the 35 on the right. Fill the (in this case 2 x 2) table with the products of the digits, separating tens and units by the upward sloping diagonal line. Finally add the digits in the upward sloping diagonals, starting from bottom-right to top-left, and use carries. This gives 0, then 5 + 1 + 6 = 12, write 2, carry 1, then 3 + 1 + 0 + 1 = 5, and finally 2. The product 2520 can now be read off. This is just a mechanization of our familiar way of long multiplication. The drawback is that a new table had to be constructed for every multiplication. The idea of the Napier bones is to use a rod for the number 7 not with the two multiples needed for this example, but with <em>all</em> multiples of 7 listed from top to bottom, and similarly for all the other digits. Placing the rods for 7 and 2 next to each other, one had to select rows 3 and 5 to make the previous product. To economize on the hardware, 4 different sets of multiples were placed on the 4 sides of the rod.</p>
<p>
</p>
<p>
Napier's <em>Promptuary</em> went a step further and turned this in an actual analog computer. Each square on the rod for digit 7 is replaced by an identical copy of a 3 x 3 block that contained all the 9 possible multiples of 7 (exclude 0 and separate tens and units above and below the main diagonal of the block in a particular way). Similarly for the digit 2. Place these two strips next to each other. Select in the (1,2) block the pattern 0/6 from 2 x 3 = 6 and mask the rest in that block. Then select in the (2,2) block the pattern 1/0 from 2 x 5 = 10, and mask the rest in that block. Identical masks are repeated in every block row. Then summing the unmasked numbers in the block diagonals as before gives the product. The ideas become much clearer with graphics like for example <a href="http://history-computer.com/CalculatingTools/NapiersBones.html" target="_blank">here</a>. Of course all this generalizes to products of integers with arbitrary lengths.</p>
<p>
A final contribution from the <em>Rabdologia</em> was a discussion of "local arithmetic". Multiplication, division, square roots were all possible if the numbers are represented in basis 2. That is binary computation several centuries before the digital computer! All entries in the tables become 0 or 1 and that obviously simplifies things considerably.</p>
<p>
Some unpublished papers were passed on in the Napier clan, and were only published much later by the historian Mark Napier as <em>De Arte Logitica</em> (1839). Havil discusses them in his 6th chapter. They deal with all sorts of subjects like decimal notation, negative numbers, irrationals, long division and multiplication, the rule of three, notations for nth roots etc.</p>
<p>
In a last chapter, Havil surveys in what ways Napier's findings have influenced his successors. His rods and Promptuary have inspired many to develop analog computing devises, and of course there is obviously the slide rule as a consequence of his logarithms. On a mathematical level he triggered the decimal Briggsian logarithms, and the natural logarithms as they are known today. Also later came the link with the area under the hyperbola, the number e (known as Napier's or Euler's constant, but the e refers to Euler), the exponential and logarithmic functions, etc.</p>
<p>
In an extensive set of appendices, Havil provides additional historical, religious, and mostly mathematical background.</p>
<p>
Julian Havil has published several other books on popular mathematical subjects that were well received. I am sure this books will be the next in the sequence of successes. It is a "general mathematics" book, and secundary school mathematics will allow to understand everything in this book. However, some affinity with mathematics will increase the appreciation of the reader. The treatise on the Book of Revelation comes as a surprising side product. For most readers, Napier will stand somewhat in the shadow of other mathematical giants, not being the brightest star on the mathematical firmament, but I'm sure after reading this entertaining and enjoyable book, Napier will climb some rungs on your ladder of famous mathematicians.</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>
As promised in the title, the book describes the life, but much more the work, of the Scottish mathematician John Napier. He published a book analyzing the biblical Book of Revelation, but most of his work is mathematical. Logarithms were his main and best known achievement, but he is also famous for his "Napier bones" and his "Promptuary", that were for a long time popular aids for computation in a pre-computer era. In subsequent chapters, Havil discusses five books containing Napier's work some of which were published posthumously.</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/julian-havil" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">julian havil</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">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">9780691155708 (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">£24.95 (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">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></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/10334.html" title="Link to web page">http://press.princeton.edu/titles/10334.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/01a40" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A40</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/01-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-02</a></li></ul></span>Mon, 24 Nov 2014 12:15:16 +0000Adhemar Bultheel45845 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/john-napier-life-logarithms-and-legacy#comments