Gresham GI Special Edition Stainless Steel Tonnaeu Case White and Blue Colourway Watch G1-0001-WHT

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Christopher Wren, who died 300 years ago this year, is famed as the architect of St Paul’s Cathedral. But he was also Gresham Professor of Astronomy, and one of the founders of a society “for the promotion of Physico-Mathematicall Experimental Learning” which became the Royal Society. All logarithmic spirals are self-similar, in that they retain precisely the same shape as they grow. In nature, if we think of how plants and animals grow, if they are growing out from a central point at a fixed rate, as happens with something like a Nautilus shell, then the outer parts continue to grow while they expand out from the centre. Logarithmic spirals allow for this to happen while keeping the same shape. The spiraling makes room for new growth. The three-dimensional version of a logarithmic spiral that Wren studied is just the right solution for shells, and is achieved in nature by one side of the structure growing at a faster rate than another. By varying the parameters in the general equation for a solid logarithmic spiral, many different shell-like shapes can be created. Wren’s ideas continue to inspire. In 2021, a team at Monash University came up with a “power cone” construction generalizing the cone-to-spiral idea (and Wren is referenced extensively in their article) that gives a mathematical basis for the formation of animal teeth, horns, claws, beaks and other sharp structures.

Spiral-like shapes crop up regularly in nature. There’s a particular kind of spiral, called a logarithmic spiral that was familiar to Wren. Logarithmic spirals were first mentioned by the German artist and engraver Albrecht Durer, and studied in great detail by the mathematician Jacob Bernoulli – he gave them the name “spira mirabilis”, or “miraculous spiral”. In a logarithmic spiral, the distance r from the centre is a power of the angle we’ve moved through (or conversely the angle is a logarithm of the distance, hence the name). This means that the gap between consecutive rings of the spiral is increasing each time. One example of a logarithmic spiral, shown below, is r= 2 θ/360(where we are measuring our angles in degrees). With every complete revolution, the distance of the spiral from the origin doubles. It crosses the x -axis at 1, 2, 4, 8, 16 and so on. Discover our convictions and minds. Share yours with us and be repost in our monthly "member discussion". Gevril GV2 Gent's Scuderia Ltd Edt Swiss Multifunction Chronograph Watch with Genuine Leather Strap Robert Hooke, oil painting on board by Rita Greer, history painter, 2009, who has made the digitized version available under the Free Art Licence http://artlibre.org/licence/lal/en/. It’s available from Wikimedia https://commons.wikimedia.org/wiki/File:17_Robert_Hooke_Engineer.JPG In February 1658, mathematicians in England received a challenge from France. It read “Jean de Montfort [possibly a pseudonym for Pascal] greatly desires that those distinguished gentlemen, the Professors of Mathematics, and others in England renowned for mathematical skill, may condescend to resolve this problem”. The problem was, given an ellipse of known dimensions, and a chord of the ellipse crossing the major axis at a known point and angle, to find the lengths of the segments of that chord. Wren solved the problem, and then in return challenged the mathematicians of France to solve another problem about ellipses, which I’ll tell you about now.

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In the course of my exploration I will not simply confine myself to English or even British history, for Britain was connected to Europe and the wider world in multifarious ways during the nineteenth and early twentieth centuries. Anyone seeking an illustration of this could do worse than to cast an eye over the Table of Contents of A. N. Wilson'sThe Victorians, with its chapters on France, Germany and Italy, India, Jamaica and Africa, and its coverage of Wagner, Dostoevsky and Tolstoy. Many of the ideas, beliefs and experiences of the Victorians were shared by people in a variety of different countries, from Russia to America, Spain to Scandinavia, and were reflected in the literature and culture of the nineteenth century, up to the outbreak of the First World War. Beyond this, overseas Empire loomed ever larger in the consciousness of the Victorians, until it came to express itself in an ideology, the ideology of imperialism. How a watch keeps time is vital to how accurate its timekeeping is. Quartz movement watches are often prized for their accuracy, while a well-crafted automatic watch is a true investment piece. Prefer something vintage? Browse our collection of stunning manual watches. Ramsdens Watch Services Gresham College, Wellcome Collection, https://www.lookandlearn.com/history-images/YW011977M Attribution (CC BY 4.0) If you’d like to read more about Wren’s life, two very good places to start are Lisa Jardine’s 2002 biography On a Grander Scale, and Adrian Tinniswood’s 2001 biography His Invention so Fertile. So, Wren and Hooke’s best guess for the ideal shape of a masonry dome is a cubic curve in cross-section. They took the part of the curve y= x 3 for positive x , and rotated it around a vertical axis to create what Hooke called a “cubico-parabolical conoid”. And it’s this shape that Wren used for the middle dome, which supports the hemispherical outer dome and its central lantern. By the way, if you stand inside the cathedral and look up, you think you can see through the dome to the lantern, but in fact what you are seeing is a painting of the lantern on the base of the middle dome! In summary, the dome of St Paul’s is in fact a triple dome: a catenary inside a cubic curve inside a hemisphere. Pretty amazing, and a tour de force of Wren’s mathematical and architectural skill.

Numerous technical obstacles had to be overcome in creating a universal system of standard time. In 1872, when the first transatlantic cable, the transmission of messages revealed that Paris was half a second further away from London than had previously been thought. Trying to fix a precise difference in longitude between Paris and Berlin, engineers noted that signals were slowed by mechanical and other factors such as the 'non-instantaneity of the transmission of the electric flux'. Despite such technical problems, and overcoming a bitterly fought rearguard action by the French, who eventually abstained on the decisive motions, in 1884, delegates from 25 states met in Washington to agree on the standardization of world time. Sailors had already synchronized time using chronometers set by longitudinal measurements based on the Greenwich Meridian, reflecting British dominance of seaborne mercantile traffic, and this was the standard adopted at the Washington conference, which divided the world into 24 time-zones by longitude, treating the meridian as the zero line, dividing the Eastern from the western hemisphere.The scale of the British Empire and the dominance of British industry ensured that in 1890 nearly two-thirds of the telegraph lines in the world were owned by British companies, which controlled 156,000 kilometers of cables. But the influence of the system extended far beyond the British Empire. The growth of the new global communication networks meant, as the writer Max Nordau noted in 1892, that the simplest villager now had a wider geographical horizon than a head of government a century before. If he read a paper he 'interests himself simultaneously in the issue of a revolution in Chile, a bush-war in East Africa, a massacre in North China, a famine in Russia'. The portrait of Christopher Wren is from the National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw06939/Sir-Christopher-Wren Neither of these demonstrations have been preserved, and it’s not clear if they were mathematical proofs or the outcomes of physical experiments. However, some years later Hooke did write down in anagram form a phrase which indicates that he had determined the solution to the problem (even if he had not necessarily found a mathematical proof): it’s a catenary. A catenary is the curve made by a chain or rope allowed to hang freely between two points. Galileo had talked about this problem; he thought that to a good approximation the solution was a parabola, but it was discovered later to be a subtly different curve. Hooke found that the equations describing the forces acting on a hanging chain are equivalent to those describing the forces acting on an arch (this time not tension and gravity but compression and gravity). That would imply that the most stable, strongest shape for an arch is a catenary, but upside-down. You can make the actual curve of the arch a slightly different shape but the line of thrust is still a catenary curve, so that needs to be part of the structure of the arch. This means the shape that requires the least amount of material, the most efficient shape, is indeed a catenary. So, we now have an outer hemispherical dome with a gigantic lantern, that can’t support itself and needs some kind of internal structure. To hide that internal structure, Wren built an inner dome whose cross section is a catenary, fitting in very nicely with other elements of the internal design. So the focus of these lectures will be on identifying and analyzing six key areas of the Victorian experience, looking at them in international and global perspective: time and space, art and culture, life and death, gender and sexuality, religion and science, and empire and race. I'll try to tease out some common factors amongst all the contradictions and paradoxes, and trace their change over time. And in no area was change more startling to contemporaries than in the topic I want to deal with this evening, namely the experience of time and space. As the century progressed, people felt increasingly that they were living, as the English essayist William Rathbone Greg put it in 1875, 'without leisure and without pause - a life ofhaste'. Comparing life in the 1880s with the days of his youth half a century before, the English lawyer and historian Frederic Harrison remembered that while people seldom hurried when he was young, now 'we are whirled about, and hooted around' without cessation. 'The most salient characteristic of life in this latter portion of the 19thcentury', Greg concluded, 'is its SPEED.' Time was becoming ever more pressing. When buying a luxury watch, the brand is a key factor. Whether you're a loyal collector or looking for fashion-forward, we have a wide range of designer watches from leading brands such as Rolex, Tag Heuer, Omega and Breitling. All of our watches are individually assessed and valued by our expert buyers to ensure pristine quality. Shop by Watch Movement

Gresham introduces the latest in cutting edge watch design and construction, fusing architectural elegance with the intricacy of traditional watch making. We’ve got a huge range of 100% genuine luxury watches from leading brands such as Rolex, Tag Heuer, Omega and Breitling, all individually assessed and valued by our expert buyers. Discover our creation & achievements in a short story. Enter in our universe and become a member of the tribe.Wren’s solution of Kepler’s problem manages to relate the areas into which the semicircle must be divided to lengths of specific circle arcs. These are then equated to carefully positioned “stretched” or “prolate” cycloids – which of course Wren already knew how to find the length of, from his own earlier work. And so he was able to solve Kepler’s problem. His solution was published by John Wallis in a 1659 treatise on the cycloid (which also included Wren’s rectification of the cycloid). If your Latin is tip-top, you can give it a read: John Wallis: Tractatus duo, prior de cycloide et corporibus inde genetis: posterior, epistolaris in qua agitur de cissoide. In a 1668 letter, the English mathematician John Wallis said that although the challenge of Kepler’s problem had been issued to the French mathematicians almost a decade previously, “there is none of them have yet (that I hear of) returned any solution”. Take that, Jean de Montfort! You can play with the effects of different shaped lenses – spherical, parabolic, and hyperbolic – using Lenore Horner’s Geogebra simulation at https://www.geogebra.org/m/Ddbpxd5X

Have a designer watch you want to sell? Or, have your eyes on a particular brand and want to part exchange? Ramsdens is happy to help. Learn More About Watches Swiss Military by Hanowa Gent's Special Anniversary Limited Edition Challenger Pro Watch with Genuine Leather Strap John Wallis had shown in the 1650s how to “rectify” a logarithmic spiral, in other words how to find its length (or more properly the length of any part of it), by transforming, or “convoluting”, it into a straight line without changing the length. Wren managed to show that a version of this idea could work a dimension higher, and could be used in reverse to convolute or twist a cone into a kind of three-dimensional or solid logarithmic spiral. He suggested these spirals could be behind the growth of snail shells and seashells. And it’s since been found that this is absolutely right.This lecture is part of the seriesThe Victorians: Culture and Experience in Britain, Europe and the World 1815-1914 But what about the support for the outer dome and lantern? What Wren did there was to build a third, middle dome – and for this he wanted the strongest possible dome shape. While the catenary is optimal for an arch, that doesn’t guarantee it’s optimal for a dome. Wren and Hooke believed that the perfect shape would in fact be the positive half of the curve y= x 3 . Why did they think this? Well, we can do a bit of investigation here. It’s similar in flavour to the fact that a parabola ( y=a x 2 ) is a good approximation to a catenary. If we think about trying to find the equation of a catenary, we see that in equilibrium, the forces at every position along a hanging chain must balance. If we think about a point (x,y)on the chain, the weight Wof the section of the chain between 0 and xwill be pulling vertically downwards, the force Fexerted by the tension from the entire left-hand half of the chain will be acting horizontally to the left, and the tension Tfrom the remaining upper right-hand part of the rest of the chain will be acting upwards along the chain, at an angle of θto the horizontal. The vertical forces balance, so we get W = Tsin θ , and F=Tcos θ . That means tan θ = W F . We can make an approximation that y x =tan θas well (this would be true if we had a straight line from the origin to (x,y) , but we actually have a curve). The final step is to make another approximation, that W is proportional to x ; this would again be true if we had a straight line from the origin to (x,y) . So we get the approximation that y x =axfor some constant a , and hence that y=a x 2 , a parabola. This is a reasonable approximation and gets better the smaller the curvature. The actual general equation of a catenary curve passing through the origin is y= 1 2b ( e bx + e -bx -2 ), where bis a chosen fixed constant. There’s an infinite series we can use to calculate this expression: y= b x 2 2 + b 3 x 4 24 + b 5 x 6 720 +… (higher powers of x ). If xis small, then successive powers of xare even smaller, so the term doing all the hard work here is b x 2 2 .If we choose a= 1 2 b , we can see that the parabola matches this very closely. Right, that was the warm-up. Now think about a dome. If we try to resolve the forces this time, the weight pulling downwards at a given point will be (approximately) proportional, not to a length, but to a surface area, and so our equivalent of y xthis time is going to be proportional, approximately, to x 2 , not x . (This is all extremely rough and ready!) So we can understand why Hooke and Wren arrived at the approximation of a cubic curve, y= ax 3 , for (a cross-section of) the ideal dome. Again, the true equation has been found since then. It’s extremely complicated! There’s a series expansion of it that begins y=a( x 3 + x 7 14 + x 11 440 +…)so for small xthe cubic equation is a good approximation. We remember Christopher Wren as a great architect. But he was so much more. Today I’m going to tell you about Christopher Wren the mathematician. We’ll look at his work on curves including spirals and ellipses, and we’ll see some of the mathematics behind his most impressive architectural achievement – the dome of St Paul’s Cathedral. Don’t worry about finding the perfect watch for your budget, because our collection of luxury watches also boasts new and pre-owned watchitems with a price-match promise, meaning if you find it cheaper elsewhere, we could match it (T&Cs apply).