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. 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. 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. Yet it seems indisputable that 'Victorian' has come to stand for a particular set of values, perceptions and experiences. On the other hand, historians are deeply divided about what these were. Certainly as G. M. Trevelyan remarked half a century ago, referring obliquely to Lytton Strachey's debunking of these values: 'The period of reaction against the nineteenth century is over; the era of dispassionate historical valuation of it has begun.' And, he added, perhaps as a warning: 'the ideas and beliefs of the Victorian era...were various and mutually contradictory, and cannot be brought together under one or two glib generalizations'. Gresham introduces the latest in cutting edge watch design and construction, fusing architectural elegance with the intricacy of traditional watch making.

This course of lectures looks at the Victorians not just in Britain but in Europe and the wider world. 'Victorian' has come to stand for a particular set of values, perceptions and experiences, many of which were shared by people in a variety of different countries, from Russia to America, Spain to Scandinavia and reflected in the literature and culture of the nineteenth century, up to the outbreak of the First World War. The focus of the lectures will be on identifying and analysing six key areas of the Victorian experience, looking at them in international perspective. The lectures will be illustrated and the visual material will form a key element in the presentations. Throughout the series, we will be asking how far, in an age of growing nationalism and class conflict, the experiences of the Victorian era were common to different classes and countries across Europe and how far the political dominance of Britain, the world superpower of the day, was reflected in the spread of British culture and values to other parts of the world.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. There were two key questions people always had about curves, known as “quadrature” and “rectification”. Quadrature is finding the area under a curve. Galileo approximated the quadrature by making a cycloid out of metal and weighing it, but he didn’t know the exact formula. We don’t know for sure when he did this, but he wrote in 1640 that he’d been studying cycloids for 50 years. At any rate, it took until the 1630s for the correct solution to be found (probably first by Gilles de Roberval): if the rolling circle has area π r 2 , then the area under each cycloid arch is 3π r 2 . Very nice. But the cycloid had still not been “rectified”: this means finding its length. The first person to do this, of all the illustrious mathematicians who had studied it, was Christopher Wren. He showed that the length is another beautifully simple formula. If the rolling circle has diameter d , its circumference is πd , and each cycloid arch has length precisely 4d . (Actually, Roberval claimed to have done this first too, but he did that a lot. He only started making this claim after Wren told Pascal the result, and Wren’s proof was the first to be published, as far as I know. The general consensus at the time and since seems to be that Wren was indeed the first to rectify the cycloid.) 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. 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. The three conics, by Pbroks13, CC BY 3.0, via Wikimedia Commons https://commons.wikimedia.org/wiki/File:Conic_sections_with_plane.svg

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. Yet as I argued in my Gresham lectures last winter, what one might call the 'long Victorian era', bounded by the end of the Napoleonic War and the beginning of the First World War, does possess a certain unity and coherence, despite its various and rapidly changing nature. This was the era when Europe, and above all Britain, achieved a leadership in and dominance of the world never matched before or since. This fact alone and the spreading consciousness of it amongst the British and European populations, helped frame attitudes and beliefs in a way scarcely possible in other epochs. One of my aims in this series is to explore how this consciousness worked itself out in practice, and how and why it grew and developed. At the beginning of the nineteenth century, communication was slow, even relatively short journeys were uncertain and time-consuming, and people were dependant on the forces of nature for energy; this lecture charts the development of new modes of communication, from the railway to the radio, the telegraph to the telephone, the steamship to the motor-car and examines their efforts on perceptions of time and space. The following paper is a helpful summary of Wren’s mathematical work which gives detail of the original sources, for example the places in Wallis’s Tractatus de Cycloide where he explain’s Wren’s rectification of the cycloid and solution to Kepler’s problem. Wren the Mathematician, D.T. Whiteside, Notes & Records of the Royal Society, 15, pp107-111 (1960).Loft of Casa Batlló, designed by Antoni Gaudí, image by Francois Lagunas, CC BY-SA 3.0, via Wikimedia Commons https://en.wikipedia.org/wiki/Casa_Batll%C3%B3 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. 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. 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.

The story of the y = x 3approximation to the perfect masonry dome, and a derivation of the correct equation, is given in Hooke's Cubico-Parabolical Conoid, by Jacques Heyman, in Notes and Records of the Royal Society of London, Vol. 52, No. 1 (Jan., 1998), pp. 39-50 https://www.jstor.org/stable/532075.Keen to recapture the initiative from the British, the French government organized an International Conference on Time in 1912, which established a generally accepted system of establishing the time and signaling it round the globe. The Eiffel Tower was already transmitting Paris time by radio signals, receiving calculations of astronomical time from the Paris Observatory. At 10 a.m. on 1 July 1913, it sent the first global time-signal, directed at eight different receiving stations dotted around the world. Thus, as one French commentator boasted, Paris, 'supplanted by Greenwich as the origin of the meridians, was proclaimed the initial time centre, the watch of the universe'. The coming of wireless telegraphy had indeed signaled the death-knell for the remaining local times. There’s an excellent article by Tony Philips on the mathematics of shells at http://www.ams.org/publicoutreach/feature-column/fcarc-shell1. I created my designs in Geogebra3D, using a modified version of the general solid logarithmic spiral equation discussed in the article. Wren was educated at Oxford and later held the Savilian chair in astronomy there, as well as his Gresham professorship in London. These roles and others place him right at the heart of an exceptionally active and exciting community of scientific thinkers. The group around Gresham College included not just Wren as Gresham Professor of Astronomy but also Robert Hooke, who was Gresham Professor of Geometry at a similar time. Wren was not just a founder member of the Royal Society (which arose out of weekly meetings at Gresham beginning in November 1660) but served as its president. And he was an active contributor in meetings – if perhaps not in subscription fees, which he had to be chased to pay up. In short, he was a key contributor to the scientific and mathematical thought of the time. We can see this, not just from his own work, but by the amount he is mentioned in the writing of others, giving credit to him for certain ideas. For example, when Isaac Newton introduces the idea of a force governed by an inverse square law in his Principia Mathematica, he says that one example is the force governing the motion of the planets “as Sir Christopher Wren, Dr. Hooke, and Dr. Halley have severally observed”. Wren’s name appears seven times in the Principia. In fact, the leading architectural historian John Summerson (1904-1992) wrote that if Wren had died at thirty, he would still have been a “figure of some importance in English scientific thought, but without the word “architecture” occurring once in his biographies”. Wren’s contributions to astronomy are the subject of a lecture by the current Gresham Professor of Astronomy, Katherine Blundell, which you can watch online: today I want to explore his mathematical contributions. This, in essence, is what I propose to do in this series of six lectures, beginning today and stretching over the next few months. I'm not going to attempt a comprehensive survey of the Victorians, or offer any kind of chronological narrative, though change over time will indeed be one of my themes.

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).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 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