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The portrait of Christopher Wren is from the National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw06939/Sir-Christopher-Wren
Within major cities, tram systems, and suburban and underground railways began to speed up traffic, just as the main roads were becoming clogged with horse-drawn cabs and carriages, automobiles and omnibuses. In 1863 the world's first underground railway, the Metropolitan, opened in London, and was soon extended, but steam locomotives posed many problems, and the cut-and-cover method of construction soon ran out of roads that could be dug up, and London turned to boring deeper lines for 'tube' trains powered by electricity, the first of which was opened in 1890. Above ground, the electric tramway system devised by Werner von Siemens began running in Berlin in 1879, and soon spread to many other countries. 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. The three conics, by Pbroks13, CC BY 3.0, via Wikimedia Commons https://commons.wikimedia.org/wiki/File:Conic_sections_with_plane.svg 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'.
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. 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.
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.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.) Gresham College, Wellcome Collection, https://www.lookandlearn.com/history-images/YW011977M Attribution (CC BY 4.0) 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.
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