European Mathematical Society - 01-02
https://euro-math-soc.eu/msc-full/01-02
enThomas 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 History of the Priority Dispute between Newton and Leibniz
https://euro-math-soc.eu/review/history-priority-dispute-between-newton-and-leibniz
<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>
Priority disputes among mathematicians are from all times, but the one between Newton and Leibniz about the discovery of calculus is notorious. Many authors, and historians have written about it. Even during the lifetime of the protagonists, the Royal Society had a commission to investigate the matter. Their conclusion was that Newton was the first, but since at that time Newton was the president of the Royal Society, this conclusion may have been a bit biased.</p>
<p>
The supporters of Leibniz whose home base was Hanover were mainly from the continent, while most of the British defended their national hero. In those days mechanics, mathematics, optics, chemistry, alchemy, astronomy, and history were all part of the job of a prominent scientist. In Newton's case certainly also theology, history, and monetary politics. While Leibniz started as a lawyer and published on palaeontology. So the whole scientific (and political) community was involved.</p>
<p>
Newton introduced his fluxions inspired by physics. A fluxion is the instantaneous change in a fluent. We now say that it is the time derivative of a function of time (the fluent). The problem was that the notion of limit was still unknown, so his peers had problems with computations that used infinitesimal small (but nonzero) quantities, that seemed to vanish when appropriate and remained nonzero at other instances. This was directly connected to the construction of a tangent and what was called a quadrature, which is the computation of the area under a curve, thus what we now call an integral. Newton's great insights happened mainly during the period of the Great Plague in 1995-1667 when he retreated to Woolthorpe Manor to live with his mother. In that time he also developed his theory of gravitation, laid the foundation of classical mechanics, and explained the planetary motion. None of this was however published until much later. The mechanics were published for the first time in his <em>Philosophiæ Naturalis Principia Mathematica</em> in 1687 and two other editions in 1713 and 1726. His book <em>The method of fluxions</em> was only written in 1671 and published in 1736.</p>
<p>
Leibniz was educated as a lawyer ans got only interested in mathematics later in 1672 when he visited Paris and meets Huygens. He was mainly concerned with quadrature. The approximate length of a curve $ds$ could be considered as the hypotenuse of a rectangular triangle with sides $dx$ and $dy$. Using geometrical arguments and similarities of triangles he obtained a method to compute the quadrature of an arbitrary curve. This was around 1674, but it was not published before 1684. He used the notation $dy/dx$ for the derivative, which was conceptually much easier to work with than Newton's fluxion notation which used the dot atop the fluent variable. This of course becomes problematic for higher order derivatives. Leibniz also introduced the integral sign ∫ as a elongated 'S' for sum, that we are still using today and which is included in the title of this book by writing "Dispute" as "Di∫pute". It is clear, and generally agreed by now, that Leibniz and Newton developed their theory independently by following different methods. However in the heat of the controversy Leibniz was accused of blatant plagiarism. Strangely enough, it were not Newton and Leibniz that stood in the barricades most of the time. In fact they exchanged polite and friendly letters. It were their followers, friends, and believers who did all the fighting on the front line, although they were of course backed up and sometimes directed by the protagonists. Newton remained more on the background, but when accusations became too direct, Leibniz had no choice but to protest against an open insult by a warrior from the opposite camp.</p>
<p>
Among the historical defenders of Leibniz were Jacob and John Bernoulli. Among Newton's warriors were John Collins, John Wallis, and Nicolas Fatio de Duillier, which is called Newton's monkey by Sonar. This Fatio has put the fuse that lit the powder keg by openly accusing Leibniz of plagiarism. At a later stage John Keill became the `army commander' of the group defending Newton. Some of the problems arose because the first correspondence was not directly between Newton and Leibniz but passed via others like Henry Oldenburg, the secretary of the Royal Society, who was not a mathematician. Oldenburg was advised on matters of mathematics by Collins, an outspoken nationalist, who was naturally opposing anything that came from the continent. There were misunderstandings, half spoken truths, and hesitation to disclose results that oxygenated the fire. The war went on, even beyond the grave. Clearly the new calculus found applications, and because Leibniz's formalism was easier, his calculus was the eventual winner. In fact it caused a drop back of the English mathematical scenery. While they were at a comparable level with the mathematics on the continent when the controversy started, they were not able to keep up with the development of calculus and analysis for a while in the eighteenth to nineteenth century post-Newton era.</p>
<p>
This fight may be well known, but disputes in those days were very common among others as well. Newton and Hooke became personal enemies over a priority dispute in optics (Newton did not want to publish his <em>Opticks</em> until after Hooke died), Huygens rejected Newton's corpuscular theory of light. He also fought with Heuraet over the rectification of curves, and he quarrelled with Hooke over a clock mechanism. Newton and Flamsteed, the Astronomer Royal, were fighting over the trajectory of the Great Comet of 1680, which Newton explained with gravity. And there were other such disputes that are also described by Sonar in this book.</p>
<p>
Thomas Sonar is from Hanover and before he engaged in the study of this history, he was rather convinced that it was a good-hearted Leibniz that was the one who was maltreated and unjustly accused by a quarrelsome and short-tempered Newton and his disciples. Sonar may have started his research with the idea of defending Leibniz, when he finished the original German version of this book in 2016, his conclusion was much more mollified. Leibniz also had not always told the truth and he wasn't the saint attacked by the devil Newton. He also had his pawns in the war and used them. This conclusion becomes clear only after meticulously investigating all the original correspondence of the seventeenth century and of all the books and papers that were published about the matter. This is the most thorough discussion of the matter that has been published so far and that still remains very readable with a minimum of mathematical knowledge, hence available for a general readership. In fact Sonar starts with an elementary introduction like a modern introductory calculus book would, so that the reader should know what calculus is about, or at least grab the meaning of derivative and integral. Then he introduces the `giants on whose shoulders Newton claimed to stand': John Wallis, Isaac Barrow, Blaise Pascal, Christiaan Huygens. So we find a biography of these people, and what they did for mathematics. In retrospect it is clear that calculus was on the doorstep, and that it only took some great minds to bring it in the open.</p>
<p>
But Sonar also gives a detailed description of the political situation and events of those days in England, France, Spain, and the Netherlands. Of course these are not really essential for the mathematics, but it sketches the framework in which scientists were working. It were usually political leaders that employed the top scientists and they made the start of academies financially possible. This part of the book has several very useful timelines, and there are many beautiful pictures throughout the book. Just reading this political prequel to the main dish is already a wonderful experience. Then of course we meet both Newton and Leibniz, how they grew up and studied and how they arrived at the discovery of their new calculus. At first there is some friction (Sonar calls it a cold war) between the two, then there is a period of relaxation, but when things get published the smoldering fire becomes a real war. Sonar includes many quotes from the letters that go back and forth about the matter with precise dates of when they were written, whether it was as an impulsive reaction to a previous message or it was written only after a long time of postponing it, possibly who was the messenger, and, not unimportant, when the letters arrived. With every new player we are given his or her background and some biography.</p>
<p>
Fortunately the excellent and smoothly reading English translation comes so shortly after the German original and was done by Sonar himself with the help of Keith Morton, his Oxford thesis advisor and later by his advisor's wife Patricia Morton. I can highly recommend this book if you have just a slight interest in history and/or mathematics. Perhaps the professional mathematical historians may not find much new or innovative material, since this 'cold case' has long been settled and solved, I believe they will still enjoy reading 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 the English translation of the German original that appeared in 2016. It is a detailed analysis of this famous controversy that is brought in an easily accessible format for a general readership. It starts with a brief introduction to differentiation and integration (that can be skipped if you don't need it), then sketches the political situation in England, France, Spain and the Netherlands of the 17th century, en finally elaborates on the rise and decline of the controversy backed up by many quotations from the letters that were mailed back and forth on this matter.</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/thomas-sonar" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Thomas Sonar</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-birkh%C3%A4user-verlag" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer International Publishing / Birkhäuser Verlag</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2018</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-72561-1 (hbk); 978-3-319-72563-5 (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">137.79 € (hbk); 107.09 € (ebk)</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">576</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://www.springer.com/gp/book/9783319725611" title="Link to web page">https://www.springer.com/gp/book/9783319725611</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/01a45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A45</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/26-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">26-03</a></li></ul></span>Tue, 29 May 2018 06:18:34 +0000Adhemar Bultheel48508 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/history-priority-dispute-between-newton-and-leibniz#commentsIslamic Geometric Patterns
https://euro-math-soc.eu/review/islamic-geometric-patterns
<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>What are the mathematics behind Islamic geometric decorations? What is the essence that makes it so recognizable? One possible characterization is by pointing to the symmetry, hence group theory is what is needed to describe it. However, that may catch some of the local symmetry, which of course is part of the beauty of these designs, but it does not completely explain the overall structure as well as the finer geometric aspects of doubling and interweaving lines that define the patterns. Thus the description of the 17 wallpaper groups is not the end of the story.</p>
<p>Jay Bonner, who is a creative designer of these patterns gives here a detailed description of the underlying polygonal techniques that can be combined to form a myriad of possible designs. He comes to his conclusion by comparing the many designs that were used throughout the Islamic cultural history and by distilling from these the techniques that were possibly used. Some of the designs, and hence the assumed underlying techniques, were more popular than others or were particular for certain regions or periods. The possibilities of the more complicated ones were not always fully explored and they give rise to new original designs. After the decline of the craftsmanship of these Islamic designs, some renewed interest in the subject arose in the second half of the twentieth century. Some books were written on the mathematics of the symmetry groups used, and it became a popular subject for documentaries and picture books, but Bonner now supersedes the latter less mathematical approaches with this monumental encyclopedia. It is not only a nice picture book with over a hundred photographs of decorative art on monuments (in chapter 1), but there are also the 540 other illustrations, many of which consist of several parts that illustrate the construction and the results of the designs.</p>
<p>The first chapter starts with a quick survey of design techniques with pointers to many illustrations in subsequent chapters where a more technical discussion is given. The main objective of the chapter however is to illustrate by a chronological summary how the different techniques were used throughout the centuries of Islamic culture from the Umayyad Caliphate (7-8th century) till the Mamluk Sultanate in Egypt (13-16th century), how they evolved in Eastern Islamic countries as well as in North Africa and the Western Al-Andalus, and how the techniques were adopted in non-Muslim cultures.</p>
<p>The second chapter is a bit more technical and summarizes different classification methods. One can for example look at an underlying regular tessellation (isometric, triangular, rectangular, hexagonal), or the known plane symmetry groups can be used to classify the designs, but the method proposed by Bonner is by design methodology, and he gives arguments why the polygonal technique is probably the one that was historically most commonly used, and hence the proper way to classify. Other authors have proposed that historically different methodologies were used but there is less evidence for those proposals or they are only useful for simpler designs. The polygonal technique starts from a polygonal tessellation of the plane. Pattern lines in these polygons will define the eventual design. These pattern lines emerge at points on the edge under particular incidence angles and intersect the pattern lines from the other edges. Once the polygons are put together to form a tessellation of the plane, the global design will protrude and the underlying polygonal stratagem can be forgotten.</p>
<p>The incidence angle of the pattern lines at the midpoints of the edges can be acute median or obtuse, and there is a fourth possibility in which pattern lines start from two symmetric points on the edges. Depending on the incidence angles and the underlying polygonal pattern rotational symmetry will occur. The most common are fourfold, (with squares and 8-pointed stars), sixfold (with 3-,6-,12-, and even 24-pointed stars), or fivefold (5- and 10-pointed stars), but occasionally also sevenfold symmetry was used, and in the more complex designs we also find 11, 13-pointed stars. Usually the stars appear at the vertices of some regular polygonal grid and/or its dual.</p>
<p>The longest chapter by far is chapter three which is a thorough discussion of the polygonal technique. One possibility is to start from a tessellation of the plane that consists of one or several types of regular polygons (triangles, squares, hexagons, octagons). Sometimes one needs the systematic inclusion of an irregular polygon, which is then called a semi-regular grid. The pattern lines can be narrow or invisible like when they just delimit coloured mosaic tiles, or they can be widened or doubled. Moreover they usually do not just intersect but they form an ingeniously interweaving pattern.</p>
<p>But regular or semi-regular tilings are relatively simple and soon Bonner moves to tessellations composed of regular and irregular polygons decorated with suitable pattern lines that fit nicely together obtained by one of the four design possibilities (acute, median, obtuse, 2-point). Bonner systematically discusses the different possible symmetries that can be obtained in this way. There are two variants of the fourfold symmetry. The A version has a large and a smaller octagon and seven other polygons to tessellate. The B version has only one octagon and five other polygons, but still that leaves many possible tessellations. The fivefold system obviously involves decagons and pentagons but can also include many other convex and concave polygons. This fivefold system was very popular and Bonner discusses several variations depending on the shapes of repeat units, that are rhombi, rectangles, or hexagons, These repeat units will fill up the plane by translation. It's not a coincidence that the golden ratio appears in these designs. Sevenfold symmetry occurs is more complicated to deal with and therefore probably less frequently used. The starting point is a tetradecagon and a heptagon and pattern lines can be constructed by connecting the midpoints of edges that are <em>k</em> = 1,...,6 positions apart.</p>
<p>A second group of design methods are called non-systematic patterns by Bonner. This technique allows the construction of more enigmatic stars with 9,11,13, or 15 points. While in the previous group, a tessellation was formed using a limited set of polygons, in this group, just one characteristic polygon is used (rhombus, triangle, square, rectangle, hexagon) that tessellates the plane. The generation goes as follows. Take one of the polygons and generate at each of its vertices, equispaced radii are generated such that the incident edges of the polygon are two of them. The intersection points of the radii are used to generate a design pattern consisting of smaller polygons, and the whole design is then translated to cover the plane. Bonner describes many examples using this kind of technique, some are historical, but there are also possibilities for original designs.</p>
<p>The most complex design technique is called dual-level design. Basically one starts from a coarse level that generates a set of lines that are widened. These wide strips are decorated with a fine gain design, which is then extended to the whole plane. This gives highly complex structures of which historical examples exist. Although there are only two levels used, it has the characteristics of self-similarity and it creates possibilities for new multilevel designs. In a short final section, some ideas are given about how to apply such techniques to decorate a dome or a sphere.</p>
<p>I do realize that my previous attempt to capture the main points of the design methodologies is totally inadequate since one needs the graphics to understand them properly. You may want to look up the author's Facebook page or the website of his company, but none will match the abundance and clarity of pictures in this book.</p>
<p>In a short chapter 4 Craig Kaplan describes the software building blocks that will be needed to generate the pictures on a computer: tilings, fitting polygons together, generating patterns lines, producing rosettes, how to join widened lines or generate the weaving effects etc. There is very little mathematics here and it remains a high level description so that it will need additional computer and mathematical skills to actually produce the graphics, but it gives at least some useful guidelines.</p>
<p>The book is very carefully edited, especially the graphics are extremely nice and very informative. The only strange typo I could spot was that σ is called "delta" on page 361. The book is not written by a mathematician, nor is it written for mathematicians. It is an artistic designers (hand)book for Islamic(-like) geometric patterns. There is very little mathematics, but I am sure all mathematicians will love the beauty of the designs non the less. While reading the text, it takes a while to get used to the terminology. There is a glossary with a set of terms that are briefly explained at the end of the book, but these are necessarily short and their meaning will become only gradually more clear. When chapter one starts with a brief survey of the techniques, one is pointed to pictures in later chapters to get an idea of what is meant, but the proper explanation comes only in chapters 2 and 3, and if you are really interested how the graphics can be produced on a computed, one has to read chapter 4. Mathematicians may be used to books that are arranged in the opposite order: start with the definitions and tools and end with the applications. The (often forward) references to pictures in this book are however carefully and consistently done, so that with a lot of paging back and forth one becomes gradually familiar with the content and the ideas proposed. The book has the looks of a coffee table book, but it requires more than just casual reading to understand the design methods.</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 marvelously illustrated book about the Islamic decorative art that is immediately recognized by its geometric patterns. The possibilities of combining designs for basic patches on diverse polygonal tiling strategies leads to a wealth of different patterns, for which some classification is proposed. The first approach is mainly historical with many pictures of the actual decorations, but there are many more graphics generated by computer to illustrate the patterns and how they are generated and repeated to fill the plane.</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/jay-bonner" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Jay Bonner</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-verlag-new-york" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Springer Verlag New York</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">978-1-4419-0216-0 (hbk); 978-1-4419-0217-7 (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">116,59 € (hbk); 91,62 € (ebk)</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">620</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/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9781441902160" title="Link to web page">http://www.springer.com/gp/book/9781441902160</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/05b45" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">05B45</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/01a30" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01A30</a></li><li class="vocabulary-links field-item even"><a href="/msc-full/51-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">51-03</a></li><li class="vocabulary-links field-item odd"><a href="/msc-full/52-03" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">52-03</a></li></ul></span>Mon, 19 Mar 2018 08:44:30 +0000Adhemar Bultheel48343 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/islamic-geometric-patterns#commentsQuite Right
https://euro-math-soc.eu/review/quite-right
<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>
In this book, Biggs tells the story of how mathematics has evolved since humans started dealing with quantities and numbers. To some extend, his arguments are that much of the human urge to develop mathematical tools has its origin in socio-economic needs. And of course, much can be argued on these grounds. That includes a fair division of a slaughtered mammoth among the hunters to the cryptographic tools needed to protect our current bank transactions.</p>
<p>
In economy, much is related to measuring quantities, values and exchange rates, the computation of interest, and many other financial transactions. Therefore measuring, numismatics, and finance are extensively discussed in the text. The facts from mathematical history can be found in many other books as well, but they are of course also covered in the current one. However it is worth noting that Biggs often adds his reservations to facts and urban legends that are sometimes posited in other books with certainty.</p>
<p>
The successive chapters deal with specific aspects in the evolution and they are roughly organized chronologically. So it starts with `the unwritten story' when written language and numbers did not exist. And yet we find tallies on bones or sticks. Some of these seem to list prime numbers, but Biggs definitely claims that assuming that a notion like prime number would be known in prehistory is pure nonsense. On the other hand, money in the form of cowrie shells was indeed used and people measured lengths expressing it in terms of body parts.</p>
<p>
Writing and counting started about 4500 years ago when people lived in settlements and the first signs of written language, economic problems and mathematics emerged. The Babylonian cuneiform numbers and the Egyptian arithmetic are well known and also length, weight, area and volume were measured. However, the myth created in the 19th century that the pyramids were built using a cosmological yard and extraterrestial knowledge have been countered later by facts and re-measuring. Gold and silver were used to pay for goods and services. This made precise weighting of precious metals and their alloys was very important.</p>
<p>
True coins were issued by kings and emperors and were for example used to collect taxes. The Greek denoted numbers by letters and solved arithmetical problems using pebbles placed in geometric arrangements to obtain results about integers. Geometry was based on using only compass and straightedge and most proofs were geometric constructions. This we know from Eulid's <em>Elements</em>, the most reprinted book ever, and which was the basis for mathematical education till rather recently. But again Biggs puts Euclid in perspective. It is not even 100% sure, although very likely, that Euclid was a real person, but whether he wrote his <em>Elements</em>as we know it is hard to believe. What we know are transcripts of translations of transcripts etc that were produced many centuries later.</p>
<p>
Through the Arabs the Hindu positional number system reached Western civilization and they also introduced algebra and algorithms, although the originals are only remotely resembling their modern counterparts. Money became widespread, but it came in all types and values. For trading and tax collecting, it became more and more important to compute exchange rates, and these computations were done by professionals.</p>
<p>
By the end of the Middle Ages, computing interest rates and solving equations were the target mathematical problems and we find here the well known story of Cardano an the Tartaglia-Fior duelling competition in their race to solve the cubic equation. The formulas were important tools in a computational profession and were often kept secret. We learn about the Roman abacus (which was originally not the instrument with beads shifting on wires as we usually think of it, but it consisted pebbles arranged in certain arrangements). It was the <em>Liber Abbaci</em> by Fibonacci that promoted the Hindu system over the Roman numerals that finally prevailed and that we are using today. Also combinatorial problems were investigated and also these have Hindu origins.</p>
<p>
Mathematics now takes major steps forward with the introduction of logarithms, infinite sums, symbolic notation became usual, Descartes combined geometry and algebra, number theory matured, and calculus was initiated by Fermat and developed by Newton. Because his publication was delayed and because Leibniz had an easier notation the latter was more popular leading to a big controversy between Newton and Leibniz camps (not elaborated in this book). Probability was born from gambling problems and the analogy with financial and insurance problems. The discovery of the bell-shaped normal distribution and the law of large numbers was quite an achievement only possible because of the collection of large numbers of statistical data. In this context, Biggs also elaborates on some game theory and the Pareto optimum. Also fair voting systems or the impossibility of such a system is discussed. He concludes with more modern economic and financial models (Black-Scholes), even Shannon's information theory, and public key cryptography. The latter of course became very important now that money became information stored in bits in a digital computer.</p>
<p>
The author of course has a point that economical problems formed an incentive to develop certain mathematical tools. Hence the subtitle <em>The story of mathematics, measurement and money</em>, and indeed the reader will probably find less mathematical details and anecdotes than in many other (popular) books on the history of mathematics. On the other hand, we find much more details about the numismatic and financial history than elsewhere. The non-monetary measures are also discussed, but not as detailed as the economic and financial aspects. The text is quite accessible without manipulating a convulsively popularizing style. There is little humor, but a pun usually comes unexpectedly so that it is really funny when it takes you by surprise. The useful illustrations and schemes are in gray-scale and to the point. They are not overdone but are added where they matter.</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 an account of the history of mathematics from the perspective of economics and finance. </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/norman-biggs" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Norman Biggs</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">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">9780198753353 (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">192</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-education-and-popularization-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics Education and Popularization of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="https://global.oup.com/academic/product/quite-right-9780198753353?cc=be&amp;amp;lang=en" title="Link to web page">https://global.oup.com/academic/product/quite-right-9780198753353?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/92b02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">92B02</a></li></ul></span>Sun, 13 Mar 2016 13:59:53 +0000Adhemar Bultheel46798 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/quite-right#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#commentsNewton and the Origin of Civilization
https://euro-math-soc.eu/review/newton-and-origin-civilization
<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>
Newton's notes <em>Chronology of Ancient Kingdoms Amended</em> were published posthumously in 1728. It consists of a chronological list of dates and events ranging from 1125 BCE with pharaoh Mephres reigning over Egypt to 331 BCE with Darius the last king of Persia being slain by Alexander the Great. In six subsequent chapters he discusses the civilizations of the Greek, Egyptians, Assyrians, Babylonians and Medes; he describes the temple of Solomon and ends with the empire of the Persians. Before this 'official' publication of 1728, an abstract made by Newton had been translated into French and was published prematurely in 1725 together with critical comments by Etienne Souciet (which were added anonymously). Because of Newton's iconoclastic views on chronology, these publications unchained a vivid controversy among scientists of the 18th century. Some of them believed in Newton and his scientific methods, others contested his computations with arguments that did or did not rely on science or just on the infallibility of the Bible.</p>
<p>
The authors of this book present a thorough analysis of the notes of Newton, to explain how he came to these conclusions about ancient history. Their analysis is both broad and deep, which allows the reader to almost look into the head of Newton as he was constantly revising and improving his ideas along with his analysis of the texts and how he adapted his measurements and computations. Buchwald and Feingold largely rely on unpublished notes by Newton and many other primary sources.</p>
<p>
They start with a discussion of Newton's views on the reliability of our senses when doing experiments and how it was possible to obtain accurate results out of multiple experiments. This, and subsequent analysis of Newton's way of thinking is placed in a broad historical perspective. They start from the ancient Greek views and show how this evolved during subsequent centuries to result in the different viewpoints on the matter among Newton and his contemporary colleagues. The authors continue to sketch the ideas about chronology and how it was perceived in the 17th century in which Newton formed his own ideas. In their next chapter, they move to Newton's views on prophecies and idolatry. For Newton, the prophecies were symbols that should be interpreted and used as a starting point for computations that should ultimately result into numbers. The mythology is some kind of sublimated history because the gods refer to kings and rulers of ancient time. Another chapter is devoted to population dynamics. An active dispute was going on about how many people lived on earth before and after the deluge, and how to compute these numbers. Newton then comes to his idea that kingdoms appear only late in history. He gets precise dates by studying ancient written sources about Persia, Egypt and Greek history. From astronomical events that he can find there, he can place a star at a particular position in the zodiac. Taking into account the precession of the earth which shifts the position of the corresponding equinox over the centuries, he was able to pin a date for the event mentioned in the text. Each of these elements that led to Newton's conclusions are elaborated in a separate chapter. How Newton evaluated words and verified them against truth is illustrated with the investigation that he did as the warden of the Royal Mint. The different stories about the leaking of Newton's summary of his notes and the premature publication of the French translation <em>Abrégé de la chronologie de M. le Chevalier Isaac Newton, fait par lui-même</em> (1725) in Paris is told with all details. The authors describe in detail the initial reactions and the lively discussions that broke loose in England and in France, certainly after the publication of the full notes in 1728, shortly after Newton's death the year before.</p>
<p>
The more technical details are collected in different appendices: a very useful list of definitions and conventions and the mathematics and computations used by Newton to find the dates. The practical computations by Newton are not mentioned in the published text, but that is amply illustrated by the authors with unpublished notes by Newton. Because of the tsunami of names of persons and sites of all ages that play a role in this book, it is very practical to consult the extensive index, and of course, given the many citations and quotes, there is also an extensive list of references.</p>
<p>
Although it is not necessary, it certainly helps to be familiar with the books of the Old Testament, and with ancient kingdoms and the Greek mythology. If not, you will probably need a great deal of wikipedia look-ups. The Chronology and many other of Newton's publications are freely available for example via the project <a href="http://www.gutenberg.org/files/15784/15784-h/15784-h.htm" target="_blank"><em>Gutenberg</em></a> as well as via the project <a href="http://www.newtonproject.sussex.ac.uk/catalogue/viewcat.php?id=THEM00183" target="_blank"><em>Newton</em></a>.</p>
<p>
With this book Buchwald and Feingold have provided the specialists with an overwhelming source of information. But anyone interested in history, i.e., ancient history but also the history up to the 17th century and before, will love the insightful discussion of ideas and the overwhelming stream of detailed facts and citations. The reader is taken by the hand and is guided through Newton's life and how his ideas have matured. A glimpse in the mind of a genius. The style and vocabulary used is not the simplest, but nevertheless it reads fluently. A highly recommended reading indeed.</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">A. Bultheel</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">KU Leuven</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>
It is explained with a thorough historical and bibliographic study how Newton arrived at his iconoclastic revision of the chronology of ancient kingdoms of Egypt, Assyria, of the Babylonians and Medes and eventually of the Persian Empire. Newton's ideas were published in his <em>Chronology of Ancient Kingdoms Amended</em> (1728). The authors illustrate how Newton applied text analysis and astronomical observations in his calculations to come to his revolutionary conclusions, as opposed to those who got their ideas solely from the (holy) scriptures.</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/jed-z-buchwald" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">jed z. buchwald</a></li><li class="vocabulary-links field-item odd"><a href="/author/mordechai-feingold" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">mordechai feingold</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">2013</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">978-069-115478-7</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">£34.95 (hardcover)</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">544</div></div></div><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/9872.html" title="Link to web page">http://press.princeton.edu/titles/9872.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/01-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-02</a></li></ul></span>Mon, 17 Dec 2012 08:14:42 +0000Adhemar Bultheel45480 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/newton-and-origin-civilization#commentsJourney through Mathematics. Creative Episodes in Its History
https://euro-math-soc.eu/review/journey-through-mathematics-creative-episodes-its-history
<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 book grew out of a mathematical history course given by the author. It has a list of 39 pages with references to historical publications which are amply cited and from which many parts are worked out in detail. This is organized in 6 chapters describing the evolution of concepts from ancient times till the 18th century to what is now generally used in calculus courses.</p>
<p>
González-Velasco has done a marvelous job by sketching this very readable historical tale. He stays as close as possible to the original way of thinking and the way of proving results. He is even using the notation and phrasing and explains how it would be experienced by scientists of those days. However at the same time he makes it quite understandable for us, readers, used to modern concepts and notation. A remarkable achievement that keeps you reading on and on. In my opinion, this is not only compulsory reading for a course on the history of mathematics, but everyone teaching a calculus course should be aware of the roots and the wonderful achievements of the mathematical giants of the past centuries. They boldly went where nobody had gone before and paved the road for what we take for granted today.</p>
<p>
What follows is a brief summary of the subjects treated in each chapter.</p>
<p>
The first chapter on <em>trigonometry</em> starts with the Greek, the Indian, and the Islamic roots (mostly geometric) of trigonometric concepts. One has to wait till the XVIth century when trigonometric tables were produced before the term sinus was used and 2 more centuries before the notation sin, cos,... was used.</p>
<p>
The second chapter on the <em>logarithm</em> is a natural consequence of the trigonometric tables as an aid for computation. $\sin A \cdot \sin B=\frac{1}{2}[\cos(A−B)−\cos(A+B)]$ could be used to multiply numbers $x\approx \sin A$ and $y\approx\sin B$. Napier (1550-1617) and Briggs (1561-1630) worked out the concepts of the logarithm in base e and base 10. Later de St. Vincent (1584-1667) connected this to areas below (integrals of) $1/x$, and Newton (1642-1727) with infinite series, while Euler (1707-1783) generalized it to a logarithms in an arbitrary basis.</p>
<p>
<em>Complex numbers</em> are introduced in chapter 3. This is tied up with the solution of a cubic equation (Cardano, 1501-1576) as square roots of negative numbers. Bombelli (1526-1572) described complex arithmetic and Euler even studied the logarithm of complex numbers, but it was only Wallis (1616-1703) and Wessel (1745-1818) who gave the geometric interpretation and made complex numbers accepted (if you can draw them, they must exist). To Hamilton (1805-1865) they were a couple of real numbers and Gauss (1777-1855) introduced the letter i for $\sqrt{-1}$.</p>
<p>
Next chapter treats <em>infinite series</em>. Summation of numerical sequences was known to the Egyptians and the Greek, but it was Leibniz (1646-1716) who first summed the inverse of the triangular numbers $1/k(k−1)$ and Euler computed $\sum 1/k^2 = π^2/6$. As for function expansions, the Indians knew a series for $\sin(x)$ in the XIVth century, but in Europe one had to wait till the XVII-XVIIIth for Newton and Euler. However it was Gregory (1638-1675) with his polynomial interpolation formulas who later inspired Taylor (1685-1731) and Maclaurin (1698-1746) to develop their well known series.</p>
<p>
Chapter 5 about <em>calculus</em> is the major part (about a quarter) of this book. Fermat (1601-1665), Gregory and Barrow (1630-1677) contributed but of course Newton and Leibniz are the main players here with the well known dispute of plagiarism as a consequence. Here the author gives a careful and detailed analysis of their contributions and concludes that they worked independently, but that Leibniz's publications lacked clarity, which made him difficult to understand for his contemporaries.</p>
<p>
The last chapter is about <em>convergence</em>. Leibniz and Newton's ideas were still rather geometric: derivatives were tangents and integrals were quadratures, thus essentially finite. The notion of limit was lacking, which was only developed later in contributions by Fourier (1768–1830), Bolzano (1781-1848), Cauchy (1789-1857), Dirichlet (1805-1859), and others. This chapter has also a remakable original section on the less known Portugese mathematician da Cunha (1744-1787) whose much earlier contribution went largely unnoticed.</p>
<p>
</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">A. Bultheel</div></div></div><div class="field field-name-field-review-legacy-affiliation field-type-text field-label-inline clearfix"><div class="field-label">Affiliation: </div><div class="field-items"><div class="field-item even">KU Leuven</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 book describes the history of mathematics that gave rise to our modern concepts in calculus: trigonometry, logarithms, complex numbers, infinite series, differentiation and integration, and convergence (limits).</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/enrique-gonz%C3%A1lez-velasco" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">enrique a. gonzález-velasco</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" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">springer</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">2011</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">978-0-387-92153-2</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">hardcover 59,95 € (net) </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">477</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/history-mathematics" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">History of Mathematics</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/book/978-0-387-92153-2" title="Link to web page">http://www.springer.com/book/978-0-387-92153-2</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/01-history-and-biography" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01 History and biography</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/01-02" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">01-02</a></li></ul></span>Fri, 09 Mar 2012 07:47:41 +0000Adhemar Bultheel45448 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/journey-through-mathematics-creative-episodes-its-history#comments