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Oxford, United Kingdom

Wall J.,Merton College
Bioethics | Year: 2015

I am sceptical as to the contribution that human rights can make to our evaluation of medical law. I will argue here that viewing medical law through a human rights framework provides no greater clarity, insight or focus. If anything, human rights reasoning clouds any bioethical or evaluative analysis. In Section 1 of this article, I outline the general structure of human rights reasoning. I will describe human rights reasoning as (a) reasoning from rights that each person has 'by virtue of their humanity', (b) reasoning from rights that provide 'hard to defeat' reasons for action and (c) reasoning from abstract norms to specified duties. I will then argue in Section 2 that, unless we (a) re-conceive of human rights as narrow categories of liberties, it becomes (b) necessary for our human rights reasoning to gauge the normative force of each claim or liberty. When we apply this approach to disputes in medical law, we (in the best case scenario) end up (c) 'looking straight through' the human right to the (disagreement about) values and features that each person has by virtue of their humanity. © 2014 John Wiley & Sons Ltd.

Stevens B.,Merton College
Archive for Rational Mechanics and Analysis | Year: 2016

Assume we start with an initial vortex-sheet configuration which consists of two inviscid fluids with density bounded below flowing smoothly past each other, where a strictly positive fixed coefficient of surface tension produces a surface tension force across the common interface, balanced by the pressure jump. We model the fluids by the compressible Euler equations in three space dimensions with a very general equation of state relating the pressure, entropy and density such that the sound speed is positive. We prove that, for a short time, there exists a unique solution of the equations with the same structure. The mathematical approach consists of introducing a carefully chosen artificial viscosity-type regularisation which allows one to linearise the system so as to obtain a collection of transport equations for the entropy, pressure and curl together with a parabolic-type equation for the velocity which becomes fairly standard after rotating the velocity according to the interface normal. We prove a high order energy estimate for the non-linear equations that is independent of the artificial viscosity parameter which allows us to send it to zero. This approach loosely follows that introduced by Shkoller et al. in the setting of a compressible liquid-vacuum interface. Although already considered by Coutand et al. [10] and Lindblad [17], we also make some brief comments on the case of a compressible liquid-vacuum interface, which is obtained from the vortex sheets problem by replacing one of the fluids by vacuum, where it is possible to obtain a structural stability result even without surface tension. © 2016 The Author(s)

Regardless of the time period, it seems the skies have always held humanity’s attention. The Babylonians were no different. Previously, science historians assumed these ancient astronomers utilized arithmetical methods to track the movement of the stars and planets. New research published in Science, however, points out that Babylonian astronomers working at least between 350 and 50 BCE were employing sophisticated geometric methods, techniques which historians previously believed did not arise until the 14th century. “The idea of computing a body’s displacement as an area in time-velocity is usually traced back to 14th century Europe,” writes Prof. Mathieu Ossendrijver, a science historian at Berlin’s Humboldt Univ. “I show that in four ancient Babylonian cuneiform tablets, Jupiter’s displacement along the ecliptic is computed as an area of a trapezoidal figure obtained by drawing its daily displacement against time. This interpretation is prompted by a newly discovered tablet on which the same computation is presented in an equivalent arithmetical formulation.” For years, Ossendrijver traveled to the British Museum to peruse their catalogue of Babylonian cuneiform tablets. The historian focused specifically on four tablets unearthed during the 19th century, which presented astronomical calculations and instructions for constructing a trapezoidal figure. Ossendrijver, knowing of the Babylonian’s affinity for Jupiter due to its association with the deity Marduk, attempted to figure out if the trapezoidal figure somehow related to the gas giant. A breakthrough came in 2014, when Assyriologist Hermann Hunger presented Ossendrijver with an old photograph of an uncatalogued tablet from the British Museum. Ossendrijver realized the computation on the tablet in the photograph matched those in the trapezoid descriptions. Comparing all the tablets, Ossendrijver discovered the Jupiter connection. The computation covered a 60-day period that started when Jupiter became visible as a morning star. “The crucial new insight provided by the new tablet without the geometrical figure is that Jupiter’s velocity decreases linearly within the 60 days. Because of the linear decrease a trapezoidal figure emerges if one draws the velocity against time,” said Ossendrijver. “It is this trapezoidal figure of which the area is computed on the other four tablets.” The astronomers also calculated when Jupiter reached the halfway point by dividing the trapezoid into two equal areas. According to Ossendrijver, similar techniques to the Babylonian method inscribed in cuneiform cropped up in the 14th century when mathematicians at Oxford’s Merton College developed the “Mertonian mean speed theorem.”

A clay tablet dating from 350 to 50 BC. REUTERS/Trustees of the British Museum/Mathieu Ossendrijver More WASHINGTON (Reuters) - Ancient Babylonian astronomers were way ahead of their time, using sophisticated geometric techniques that until now had been considered an achievement of medieval European scholars. That is the finding of a study published on Thursday that analyzed four clay tablets dating from 350 to 50 BC featuring the wedge-shaped ancient Babylonian cuneiform script describing how to track the planet Jupiter's path across the sky. "No one expected this," said Mathieu Ossendrijver, a professor of history of ancient science at Humboldt University in Berlin, noting that the methods delineated in the tablets were so advanced that they foreshadowed the development of calculus. "This kind of understanding of the connection between velocity, time and distance was thought to have emerged only around 1350 AD," Ossendrijver added. The methods were similar to those employed by 14th century scholars at University of Oxford's Merton College, he said. Babylon was an important city in ancient Mesopotamia, located in Iraq about 60 miles (100 kilometers) south of Baghdad. Jupiter was associated with Marduk, the city's patron god. Babylonian astronomers produced tables with computed positions of the planets, Ossendrijver said. "They provided positions needed for making horoscopes ordered by clients, and they also held the view that everything on Earth - from river levels to market prices, for example grain, and weather - is connected to the motion of the planets. So by predicting the latter they hoped to be able to predict things on Earth," Ossendrijver added. He noted that the tablets themselves do not mention anything about these astrological applications. The four tablets, excavated around 1880, were stored at the British Museum in London. The cuneiform characters were impressed in soft clay with a reed stylus and the tablets may have been stored in the scientific library of an astronomer or a temple building, Ossendrijver said. The tablets contain geometrical calculations based on a trapezoid's area, and its long and short sides. It had been thought that Babylonian astronomers relied only on arithmetical concepts, not geometric ones. The ancient Greeks also were known for using geometry, but the Babylonian tablets employ it in a more complex, abstract manner. The research was published in the journal Science.

Left: Cuneiform tablet with calculations involving a trapezoid. Right: A visualization of trapezoid procedure on the tablet: The distance travelled by Jupiter after 60 days, 10º45', is computed as the area of the trapezoid. The trapezoid is then divided into two smaller ones in order to find the time (tc) in which Jupiter covers half this distance. Credit: Mathieu Ossendrijver (HU) Ancient Babylonians are now believed to have calculated the position of Jupiter using geometry. This is revealed by an analysis of three published and two unpublished cuneiform tablets from the British Museum by Prof. Mathieu Ossendrijver, historian of science of the Humboldt-Universität zu Berlin. The tablets date from the period between 350 and 50 BCE. Historians of science have thus far assumed that geometrical computations of the kind found on these tablets were first carried out in the 14th century. Moreover, it was assumed that Babylonian astronomers used only arithmetical methods. "The new interpretation reveals that Babylonian astronomers also used geometrical methods", says Mathieu Ossendrijver. His results are published in the current issue of the journal Science. On four of these tablets, the distance covered by Jupiter is computed as the area of a figure that represents how its velocity changes with time. None of the tablets contains drawings but, as Mathieu Ossendrijver explains, the texts describe the figure of which the area is computed as a trapezoid. Two of these so-called trapezoid texts had been known since 1955, but their meaning remained unclear, even after two further tablets with these operations were discovered in recent years. One reason for this was the damaged state of the tablets, which were excavated unscientifically in Babylon, near its main temple Esagila, in the 19th century. Another reason was, that the calculations could not be connected to a particular planet. The new interpretation of the trapezoid texts was now prompted by a newly discovered, almost completely preserved fifth tablet. A colleague from Vienna who visited the Excellence Cluster TOPOI in 2014, the retired Professor of Assyriology Hermann Hunger, draw the attention of Mathieu Ossendrijver to this tablet. He presented him with an old photograph of the tablet that was made in the British Museum. The new tablet does not mention a trapezoid figure, but it does contain a computation that is mathematically equivalent to the other ones. This computations can be uniquely assigned to the planet Jupiter. With this new insight the other, thus far incomprehensible tablets could also be deciphered. In all five tablets, Jupiter's daily displacement and its total displacement along its orbit, both expressed in degrees, are described for the first 60 days after Jupiter becomes visible as a morning star. Mathieu Ossendrijver explains: "The crucial new insight provided by the new tablet without the geometrical figure is that Jupiter's velocity decreases linearly within the 60 days. Because of the linear decrease a trapezoidal figure emerges if one draws the velocity against time." "It is this trapezoidal figure of which the area is computed on the other four tablets", says the historian of science. The area of this figure is explicitly declared to be the distance travelled by Jupiter after 60 days. Moreover, the time when Jupiter covers half this distance is also calculated, by dividing the trapezoid into two smaller ones of equal area. "These computations anticipate the use of similar techniques by European scholars, but they were carried out at least 14 centuries earlier", says Ossendrijver. The so-called Oxford calculators, a group of scholastic mathematicians, who worked at Merton College, Oxford, in the 14th century, are credited with the "Mertonian mean speed theorem". This theorem yields the distance travelled by a uniformly decelerating body, corresponding to the modern formula S=t•(u+v)/2, where u and v are the initial and final velocities. In the same century Nicole Oresme, a bishop and scholastic philosopher in Paris, devised graphical methods that enabled him to prove this relation. He computed S as the area of a trapezoid of width t and heights u and v. The Babylonian trapezoid procedures can be viewed as a concrete examples of the same computation. Furthermore, it was hitherto assumed that the astronomers in Babylon used arithmetical methods but no geometrical ones, even though they were common in Babylonian mathematics since 1800 BCE. Ancient Greek astronomers from the time between 350 BCE and 150 CE are also known for their use of geometrical methods. However, the Babylonian trapezoid texts are distinct from the geometrical calculations of their Greek colleagues. The trapezoid figures do not describe configurations in a real space, but they come about by drawing the velocity of the planet against time. As opposed to the geometrical constructions of the Greek astronomers the Babylonian trapezoid figures exist in an abstract mathematical space, defined by time on the x-axis and velocity on the y-axis. More information: M. Ossendrijver. Ancient Babylonian astronomers calculated Jupiters position from the area under a time-velocity graph, Science (2016). DOI: 10.1126/science.aad8085

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