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Extensible Markup Language | 1995-08-15 | 13KB | 29 lines

<?xml version="1.0" encoding="utf-8" ?> <!DOCTYPE card PUBLIC "-//Apple, Inc.//DTD card V 2.0//EN" "" > <card> <id>42595</id> <filler1>0</filler1> <cantDelete> <false /> </cantDelete> <showPict> <true /> </showPict> <dontSearch> <false /> </dontSearch> <owner>5472</owner> <link rel="stylesheet" type="text/css" href="stylesheet_3106.css" /> <content> <layer>background</layer> <id>25</id> <text>athematics and its Applications (4 of 4)In triangle DEF, DE = EF = 1, so the angles at D and F are equal, that is they are each 45 deg (the internal angles of a triangle add up to 180 deg ). Using Pythagoras' theorem, DF' = 1' + 1' = 2, so DF = 2. We can therefore conclude: 1sin 45 deg = --- ΓêÜ 2 1cos 45 deg = --- ΓêÜ 2tan 45 deg = 1In triangle GHK, GH = HK = KG = 2, so the angles at G, H and K are equal, that is they are each 60 deg . Using Pythagoras' theorem, KL' + 1' = 2', so KL = 3. We therefore have:sin 60 deg = ┬┤1/2 ΓêÜ3 = cos 30 deg cos 60 deg = 1/2 = sin 30 deg tan 60 deg = ΓêÜ 3tan 30 deg = 1/ΓêÜ 3EJBSOME EMINENT MATHEMATICIANSPythagoras (c. 582-500 BC), Greek philosopher. Born in Samos, he founded a religious community at Croton in southern Italy. The Pythagorean brotherhood saw mystical significance in the idea of number. Popularly remembered today for Pythagoras' theorem.Euclid (c. 3rd century BC), Greek mathematician. Euclid devised the first axiomatic treatment of geometry and studied irrational numbers. Until recent times, most elementary geometry textbooks were little more than versions of Euclid's great book.Archimedes (c. 287-212 BC), Greek mathematician, philosopher and engineer, born in Syracuse, Sicily. His extensions of the work of Euclid especially concerned the surface and volume of the sphere and the study of other solid shapes. His methods anticipated the fundamentals of integral calculus.Descartes, Ren├⌐ (1596-1650), French philosopher, mathematician and military scientist. Descartes sought an axiomatic treatment of all knowledge, and is known for his doctrine that all knowledge can be derived from the one certainty: Cogito ergo sum (`I think therefore I am'). One of his major mathematical contributions was the development of analytical geometry, whereby geometrical figures can be described in algebraic terms.Newton, Sir Isaac (1643-1727), English mathematician, astronomer and physicist. Newton came to be recognized as the most influential scientist of all time. He developed differential calculus and his treatments of gravity and motion form the basis of much applied mathematics.Euler, Leonhard (1707-83), Swiss-born mathematician, who worked mainly in Berlin and St Petersburg. He was particularly famed for being able to perform complex calculations in his head, and so was able to go on working after he went blind. He worked in almost all branches of mathematics and made particular contributions to analytical geometry, trigonometry and calculus, and thus to the unification of mathematics. Euler was responsible for much of modern mathematical notation.Gauss, Carl Friedrich (1777-1855), German mathematician. He developed the theory of complex numbers. He was director of the astronomical observatory at G├╢ttingen and conducted a survey, based on trigonometric techniques, of the kingdom of Hanover. He published works in many fields, including the application of mathematics to electrostatics and electrodynamics.Cauchy, Baron Augustin-Louis (1789-1857), French mathematician and physicist. He developed the modern treatment of calculus and also the theory of functions , as well as introducing rigor to much of mathematics. As an engineer he contributed to Napoleon's preparations to invade Britain, and he twice gave up academic posts to serve the exiled Charles X.Boole, George (1815-64), English mathematician. Despite being largely self-taught, Boole became Professor of Mathematics at Cork University. He laid the foundations of Boolean algebra, which was fundamental to the development of the digital electronic computer.Cantor, Georg (1845-1918), Russian-born mathematician who spent most of his life in Germany. His most important work was on finite and infinite sets. He was greatly interested in theology and philosophy.Klein, Christian Felix (1849-1925), German mathematician. Klein introduced a program for the classification of geometry in terms of group theory. His interest in topology (the study of geometric figures that are subjected to deformations) produced the first description of a Klein bottle - which has a continuous one-sided surface.Hilbert, David (1862-1943), German mathematician. In 1901, Hilbert listed 23 major unsolved problems in mathematics, many of which still remain unsolved. His work contributed to the rigor and unity of modern mathematics and to the development of the theory of computability.Russell, Lord Bertrand (1872-1970), English philosopher and mathematician. Russell did much of the basic work on mathematical logic and the foundations of mathematics. He found the paradox now named after him in the theory of sets proposed by the German logician Gottlob Frege (1848-1925), and went on to develop the whole of arithmetic in terms of pure logic. He was jailed for his pacifist activities in World War 1. In 1950, he was awarded the Nobel Prize for Literature.CHAOS THEORYFrom its beginnings, science has been a quest for orderly laws that govern nature. And with each advance it has seemed that some element of disorder has been conquered. Complex dynamical systems, in particular, could be understood and quantified when the calculus was invented. But scientists have long recognized that many natural phenomena - the movement of clouds, turbulence in streams or in the rising smoke from a cigarette, the movement of a leaf in the wind, the patterns of brain waves, disease epidemics or traffic jams - are so inherently disordered and chaotic as to seem to defy any attempt to find governing laws.As early as 1903, however, the French mathematician Jules Henri Poincar├⌐ (1854-1912) - famous for his work on topology - recognized that there are circumstances in which tiny inaccuracies in initial conditions can be multiplied so as to lead to huge differences in the outcome. Poincar├⌐'s work was largely forgotten until in 1961 the American meteorologist and mathematician Edward N. Lorenz, working with a crude early computer, set out to produce a mathematical model of how the atmosphere behaves. In the course of this work Lorenz accidentally hit upon the first mathematical system in which small changes in the initial conditions led to overwhelming differences in the outcome. Lorenz showed that this phenomenon made long-range weather prediction almost impossible. His work and the analogies that developed from it attracted the attention of scientists in other fields and led to the development of a new branch of mathematics - chaos theory. One of the most striking of these analogies is known as the `butterfly effect' - the idea that the air perturbation caused by the movement of a butterfly wing in China can cause a storm a month later in New York.By the 1970s some scientists and mathematicians, and even some economists, were beginning to investigate disorder and instability. Physiologists were considering patterns of chaos in the action of the heart-patterns that could lead to sudden cardiac arrest; electronic engineers were investigating the sometimes chaotic behavior of oscillators; ecologists were examining the seemingly random way in which wildlife populations changed; chemists were studying unexpected fluctuations in chemical reactions; and economists were wondering whether some order might be found in random stock-market price fluctuations.The first indication for an underlying pattern in chaos was found by the American physicist Mitchell Feigenbaum. In 1976 Feigenbaum noticed that when an ordered system starts to break down into chaos, it often does so in accordance with a consistent pattern in which the rate of occurrence of some event suddenly doubles over and over again. This is exactly what happens in fractal geometry - in which any part of a figure is a reduced copy of a larger part. Feigenbaum also discovered that at a certain constant number of doublings, the structure acquires a kind of stability. This numerical constant, called Feigenbaum's number, can be applied to a wide range of chaotic systems. To understand what mathematicians mean by chaos it is best to consider a simple example. Iteration is the mathematical process in which the result of a calculation is applied as the starting point for a repeat of the same process and so on. One might, for instance, take a number and halve it, then take the result and halve that, and so on repeatedly. The set of numbers that result is called the orbit of the number. Starting with, say, 16, the orbit would be 8, 4, 2, 1, 1/2, 1/4, 1/8, 1/16... Again, one might perform an iterative process on any number (x) between 0 and 1, the process being `multiply the product of x and 1 -- x by 3'. This gives a readily predictable orbit. Surprisingly, iteration for numbers between 0 and 1 using the process `multiply the product of x and 1 -- x by 4' produces a chaotic orbit for some numbers and a predictable one for others. Closely related starting values give orbital numbers that are widely different. In other words, the system is sometimes highly sensitive to its starting values, sometimes not. This is characteristic of what mathematicians mean by chaos. Chaos theory attempts to describe how such systems change from predictable to wholly disordered.Today there is much debate as to whether chaos theory, so far as it goes at present, actually does adequately describe seemingly disordered dynamical systems in nature - whether it really is, as some have claimed, a new mathematical tool of the same order of importance as calculus, or even that it is a discipline to rank in importance with relativity and quantum mechanics. The controversy rages on but the level of interest and the volume of research continue to rise. Major developments, one way or the other, are to be expected soon.</text> </content> <content> <layer>background</layer> <id>23</id> <text>ΓÇó ASTRONOMYΓÇó PHYSICSΓÇó CHEMISTRYΓÇó THE HISTORY OF SCIENCE</text> </content> <content> <layer>background</layer> <id>36</id> <text>4204060</text> </content> <name>p062-4</name> <script></script> </card>