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      Duncan Graham-Rowe
       
      Astronomers are taking the search for somewhere quiet to work
      to new extremes with a plan to put a radio telescope on the far side of the
      Moon. The advantage of this unusual location is that the
      Moon would act as a massive shield, protecting the telescope against
      radio emissions from Earth. Astronomers could also listen to low radio
      frequencies that don't penetrate the Earth's atmosphere. Claudio Maccone,
      an astronomer at the Centre for Astrodynamics in Turin, Italy, is assessing
      the concept for the International Academy of Astronautics. He even has his
      eye on a plot of lunar real estate. A 100-kilometre-wide crater called
      Daedalus should provide enough space, he says. The crater's
      3-kilometre-high rim should also help block any stray radio signals that
      creep around the Moon to the far side. "I do believe this will be built,"
      says Maccone, although he admits it will probably take at least 15 years.
      Even if robots were used to build the observatory remotely, it would cost
      billions of dollars and need the backing of a large space agency like NASA
      or the European Space Agency. By the time the telescope could be built, the
      area of the Moon that's protected from radio waves is likely to be shrinking
      fast. This is because as orbit space for telecommunications satellites gets
      used up, they will have to be placed in higher orbits, so their radio emissions
      will reach more and more of the Moon's surface (see Graphic). So Maccone
      also wants to give the region around the Daedalus crater some form of protection
      status, to create a permanent quiet zone that would be safe no matter what
      technology is developed in the future. "The far side is in my opinion a unique
      treasure that should be preserved for the sake of humankind," he says. Setting
      up such a zone would probably be the responsibility of the International
      Telecommunications Union, which allocates the rights to use different radio
      frequencies. But it's far from clear whose permission would be needed to
      build a permanent structure on
      the Moon. Maccone is
      due to present the results of his study to the International Astronautical
      Congress next October. If the plans are approved, the first step will be
      to design a satellite probe to orbit the Moon and check there really is a
      quiet zone.  
      
       
       
       
	  
      
	
	  
	    Jupiter's giant light show  | 
	 
	
	  
	     SOMETHING
	    strange is happening on Jupiter. Its magnetic field extends hundreds of times
	    further out into space than previously thought, creating auroras that make
	    the Earth's northern lights seem feeble in comparison. Jupiter is the giant
	    of the Solar System, more than a thousand times as massive as Earth. In January
	    2001, the combined power of the Cassini and Galileo space probes, the Chandra
	    X-ray telescope and the Hubble Space Telescope were all trained on the Jovian
	    magnetosphere - the region controlled by the planet's magnetic field. Magnetic
	    field lines fan out from a planet like the lines of iron filings from the
	    poles of a bar magnet. Auroras are caused by ions zipping along these lines,
	    so researchers can use the location of auroras to track how far out into
	    space the planet's magnetic field lines can trap ions from the solar wind.
	     
	    Randy Gladstone and his colleagues at the Southwest Research Institute in
	    San Antonio, Texas, used Chandra to map the Jovian auroras. Earth's northern
	    lights shine only with visible light, but the more violent Jovian auroras
	    emit X-rays.  
	    The X-ray auroras on Jupiter extend surprisingly far from the planet's poles,
	    showing that field lines reach far out into space. Gladstone also found that
	    the auroras pulsated regularly every 45 minutes in certain places he's calling
	    "hot spots", unlike anything seen on Earth. "Those field lines go way further
	    out than expected," says Gladstone. "Something weird is happening."  
	    Theorists have trouble explaining why Jupiter's magnetosphere is so much
	    more powerful than Earth's, even allowing for the planet's greater size.
	    "Jupiter's magnetosphere is like Earth's on steroids," says Thomas Hill,
	    who works on the theory of magnetospheres at Rice University in Houston,
	    Texas. Eugenie Samuel, Boston More at. Nature (vol 415, p 1000) 
	    [New Scientist 2 March 2002]
	    
	     HOW JUPITER GOT
	    ITS STRIPES. A new study of turbulence in the atmosphere around a rotating
	    sphere is helping to explain the dramatic stripes on Jupiter, Saturn, and
	    the other giant planets. On Earth, turbulence caused by solar heating and
	    friction with the ground disrupts atmospheric flows and dissipates the energy
	    provided by the sun that might otherwise lead to the formation of circulating,
	    global cloud bands. In the thin atmospheres of gas giants, however, energy
	    dissipation is small, and some of the sun's energy is gradually collected
	    in stable, global jets that trap clouds and form planetary stripes. Researchers
	    at the University of South Florida and Ben-Gurion University of the Negev
	    (Israel) have now developed a model that shows how planetary rotation and
	    nearly two-dimensional atmospheric turbulence may combine to create large
	    scale structures. Scientists have long suspected that the interaction between
	    planetary rotation and large-scale turbulence governs the banded circulations
	    on giant planets. The new research has quantified the phenomenon, leading
	    to an equation that characterizes the distribution of energy among different
	    scales of motion, and to simple formulae that describe basic energetic features
	    of giant planets' circulations. The model helps explain the paradoxical
	    observation that the outer planets have stronger atmospheric flows, even
	    though the energy provided by the sun to maintain such flows decreases with
	    increasing distance from the sun. The researchers (B. Galperin,
	    [email protected], 727-553-1101) have found that the atmospheres of
	    distant planets dissipate even less energy than their warmer sisters. Although
	    the outer planets receive less energy from the sun, they keep more of the
	    energy they receive. As a result, the model shows why Neptune has the strongest
	    atmospheric circulation of all the gas giants even though it is the farthest
	    of the bunch from the sun. (S. Sukoriansky, B. Galperin, N. Dikovskaya, Physical
	    Review Letters, 16 September 2002)  
	      | 
	 
       
      
       
	
	  
	    Mercury and the mystery of a "planet" that disappeared  | 
	 
	
	  Experience has taught me that predictions
	    of spectacular astronomical events are tantamount to long-range
	    forecasts of bad
	    weather: the grand alignment of five planets scheduled
	    to take place over the western horizon tomorrow evening therefore means that
	    gardeners should put any house plants needing a watering outside at about
	    8pm.  
	    The various planets, fortunately, will be dancing around each other in the
	    same part of the sky for a few weeks, so I still hope for a glimpse of Mercury,
	    the one planet visible to the naked eye that I have yet to see. Tomorrow's
	    planetary alignment will form a vast celestial finger pointing to its location
	    at the bottom of a line that baa Jupiter at its top, followed by Saturn,
	    Mars and Venus, with the Moon just below the Red Planet.  
	    One consequence of Mercury's elusiveness is that astronomers know little
	    about this planet. Data sent back by the only space probe to visit it,
	    Mariner 10 in 1974, raised as many mysteries as it solved. The mass
	    of the planet turned out to be extraordinarily high: despite being not much
	    larger than the Moon, Mercury proved to be five times more massive - suggesting
	    that its rocky, cratered surface covers a colossal nickel-iron ball more
	    than 2,000 miles across.  
	    The planet also has a relatively strong magnetic field, thus cocking a snook
	    at scientists who claim to know how such fields are generated. Standard theories
	    link them to the spin of the planets, which creates a kind of
	    dynamo effect within electrically conducting
	    liquid cores. Mercury's metal core is certainly conducting, but it isn't
	    liquid and the planet takes a leisurely 58 days to spin once on its axis
	    (which, mysteriously, is precisely two thirds the time Mercury takes to orbit
	    the Sun).  
	    The answers to such puzzles almost certainly lie
	    in the proximity of Mercury to the Sun. There is no object' however, similar
	    to Mercury hy which astronomers can test their theories. Not that this has
	    prevented some of the more imaginative from suggesting that Mercury may not
	    be the only object to inhabit the searing inner reaches of our
	    solar system.  
	    During the 1840s, astronomers were struggling to understand Mercury's slow
	    pirouettes around the Sun. Most of the rotation could be explained by known
	    effects, but a tiny discrepancy remained. The mystery was tackled by Urbain
	    Leverrier, the brilliant French theoretician who had in 1846 explained
	    discrepancies in the orbit of Uranus as being due to an undiscovered planet,
	    subsequently confirmed as Neptune. In 1859 Leverrier proposed that Mercury
	    was being pulled off course by another undiscovered planet, duly named Vulcan.
	     
	    Shortly afterwards, Leverrier received a letter from a French country doctor
	    claiming to have seen Vulcan crossing the face of the Sun in March 1859.
	    Calculating an orbit for the object, Laverrier found that it must lie just
	    13 million miles from the Sun. This neatly explained why no one had seen
	    it: Vulcan was even smaller than Mercury and would normally be lost in the
	    Sun's glare.  
	    Not unreasonably, Leverrier went to his grave believing he had predicted
	    the existence of two new planets. He proved to be only half right. Vulcan
	    was never seen again, and the real explanation for Mercury's anomalous orbit
	    turned out to be general relativistic effects unimagined in Leverrier's time.
	     
	    Vulcan may now reside in the wastebin of scientific history, but it is odd
	    that the early claims to have seen it should have been so consistent. That
	    said, readers who think they see Vulcan over the weeks ahead should probably
	    take it as a reminder to have an eye test. [Sunday Telegraph April 14
	    2002] 
	     
	    Robert Matthews
	    invites you to send. in your questions on science, the answer's to which
	    will soon form a regular part of his column whether you're stumped by celestial
	    mechanics or just want to know why the sea is salty, please write to Robert
	    Matthews at The Sunday Telegraph, 1 Canada Square, London, E14 5DT, or e-mail:
	    [email protected]
	     | 
	 
       
      
       
       
       
	
	  The zodiac
	    can at times appear predominantly masculine as most of the planets have
	    historically been associated with male mythological figures. Until the discovery
	    of the asteroid belt in the 19th century, the only planetary bodies which
	    women could easily identify themselves with were
	    Venus, the Moon and Lilith (which is considered to be Earth's ante-diluvian
	    satellite).  
	    While women have always made significant contributions to society, it must
	    be stressed that they traditionally played the roles of wife and mother as
	    illustrated by Venus and the Moon respectively. The discovery of the asteroid
	    belt brought about a balance between masculine and feminine archetypes to
	    bring more sexual equality in our Solar system.
	     
	    It was also around this time that women began to take on more diversified
	    roles and therefore needed other
	    astrological
	    indicators in what was to become a rapidly changing world. It is also
	    worth noting that those men who have chosen to pursue careers which have
	    traditionally been assigned to women will be able to relate to these more
	    intuitive planetary bodies.  
	     
	    The Moon 
	    The Moon is the Earth's largest satellite which governs our
	    emotional responses. It is no secret that the close
	    proximity of the Moon to the Earth has a strong effect on our physical
	    environment, as evidenced by the daily motion of oceanic tides and the fact
	    that crime has a tendency to increase during the Full Moon.  
	    The Moon relates to our emotive impulses which push us to react and in many
	    instances it can be such a dominant feature in some horoscopes that it is
	    actually more prominent than the Sun. This may
	    be, as many whose natal Moon is strongly aspected in their natal chart will
	    find to be the case, the reason why some do not show a strong affinity to
	    their natal Sun but instead relate more to Lunar impressions.  
	    The Moon represents the relationship we have with our mother and women
	    in general and it denotes what type of mother or parent we are likely
	    to become. While the father is quite often associated with Saturn, those
	    men who are more in tune with the intuitive side of themselves can just as
	    easily identify with the Moon in relation to their parenting style. In Greek
	    mythology, the Moon is associated with Artemis, the virgin goddess. She was
	    considered to be eternally young and active and was equal to any man. Artemis
	    was the twin sister of Apollo who astrologically depicts the Sun and the
	    pair were said to have been very close. 
	    With these mythological considerations in mind, it should come as no surprise
	    to conclude that the astrological signs of Leo and Cancer, ruled by the Sun
	    and Moon respectively, reign side by side in the zodiac while the remaining
	    five planets in the ancient Solar system progress outwardly from the
	    royal pair.  
	    The planetary alignment is as follows: Mercury, the first planet away from
	    the Sun, rules Gemini and Virgo which are one sign away from the Cancer/Leo
	    configuration respectively. Venus, the second planet away from the Sun, rules
	    Taurus and Libra which are two signs away from the Cancer/Leo configuration
	    respectively. Mars, the third planet from the Sun excluding Earth, rules
	    Aries and Scorpio which are three signs away from the Cancer/Leo configuration
	    respectively. Jupiter, the fourth planet from the Sun excluding Earth, rules
	    Pisces and Sagittarius which are four signs away from the Cancer/Leo
	    configuration respectively. Saturn, the fifth planet from the Sun excluding
	    Earth, rules Capricorn and Aquarius which are five signs away from the Cancer/Leo
	    configuration respectively. 
	    The placement of these signs and the rulerships they were given some 2,000
	    years ago does seem to suggest that the Summer months were regarded as
	    significant in the astrological year.[Predictions magazine July
	    1997] | 
	 
       
      
       
	
	   Question: What is a blue moon and what is once in a blue moon?
	    The three most popular reasons that are most often found in the literature
	    for this so called phenomena are 1) The appearance of the moon's color due
	    to smoke particles from forest fires, 2) The appearance of the moon's color
	    due to particles from volcanic eruptions, or 3) The appearance of a second
	    full moon within a calendar month. Apparently, both 1 and 2 have been known
	    to give the moon a blue appearance. The third is the source of the popular
	    statement "once in a blue moon" which means something happens quite rarely.
	    It is based on the strange occurrence of having two full moons in one month.
	    How can this happen? Since full moons actually occur every about 29.53 days,
	    they occur about 12.36 times a year, which translates into having a double
	    full moon month once every 2.77 years or so, quite infrequently, obviously.
	    With the event occurring approximately every 33 months, it is easy to see
	    how the statement "once in a blue moon" could have originated. It might also
	    just be the case that some specific second full moons in months past did,
	    in fact, occur right after a forest fire or volcanic eruption, providing
	    some credibility to all three. It is interesting to note that in a year where
	    a full moon does not occur in February, both January and March will have
	    two full moons, a rare event indeed.  
	    Prepared by AACTchWill and AACTutorNY, AAC Staff, Use of this material
	    is protected under America Online and other copyright. Any use of this material
	    must cite AOL's Academic Assistance Center and the authors as a source. (edited
	    by AACDrAnne) | 
	 
       
      
       
	
	  
	      The Golden age of Brum
	    
	    
	     
	      
		  | 
	       
	      
		| A subject for discussion at the Lunar Society A 1749
		  experiment using boys to transmit electricity | 
	       
	     
	    
	     John Adamson praises the elegant study of the friends who
	    meet once a month to change the world [The Lunar Men:The Friends who
	    made the future by Jenny Uglow Faber £25,518pp] 
	    EXCEPTIONAL times, in fortunate places, nature and nurture combine to produce
	    a superabundance of talent that is wholly out of scale with what can reasonably
	    be expected of mortal men. Historical moments such as the 5th century BC
	    in Athens or the later 15th century Florence have long been regarded with
	    reverential awe. This new book makes a compelling case for adding another
	    - one that, hitherto, has been relatively overlooked. The time is the second
	    half of, the 18th century, and the place -improbably - Birmingham, then a
	    semi-rustic town with a population under 50,000 people.  
	    Between the 1760s and the early 1800s, there clustered here one of history's
	    most exceptionally gifted networks of friends. Their energy, ideas and
	    inquisitiveness "made the future" - to use this book's fully justified subtitle:
	    from the development of the steam engine, through to the development of
	    mass-market capitalism, and even in the development of the theory of
	    evolution. 
	    These were the Lunar Men: a dozen or so luminaries who, from 1775,met together
	    as the Lunar Society of Birmingham (so named because they met monthly, on
	    the Monday closest the full moon). At its core was its genial pater familias
	    the portly Erasmus Darwin (1731-1802) - physician, botanist, best-selling
	    poet, balloonist, and grandfather of Charles. Around
	    him clustered the talented friends with whom he shared his enthusiasms: the
	    pottery tycoon Josiah Wedgwood, the silversmith Matthew Boulton, the inventor
	    James Watt, the chemist and Nonconformist divine Joseph Priestley; and a
	    series of less well-known but equally able colleagues. 
	    Indeed, no single epithet does any of them justice, as the Lunar Men were
	    polymaths all. The intellectual heirs of Newton in science and Rousseau and
	    the French Encyclopaedists in philosophy, they were buoyed by the belief
	    that every aspect of the physical world would eventually yield up its secrets
	    to well directed experiment and enquiry. Superstition (if not yet Christianity
	    as a whole) was their enemy. As Josiah Wedgwood put it mischievously,
	    their ambition was to "rob the Thunderer of his Bolts"; Erasmus Darwin revelled
	    in "the slightly blasphemous glamour of scientific
	    'miracles'". 
	    And so they experimented, annotated, compared results, drank claret and,
	    above all, talked. To read Jenny Uglow's book is to eavesdrop on conversations
	    about an astonishingly diverse array of enquiries, schemes and projects:
	    from how to get the correct glaze for Wedgwood's Portland Vase; to schemes
	    for moving manure using hot-air balloons; on to Erasmus Darwin's conclusion,
	    reached during the 1770s, that all life must have had a single microscopic
	    common ancestor: the first coherent theory of evolution.  
	    What shines through, however, in all their correspondence, is the invigorating
	    excitement of the chase, the magpie-like,
	    omnium-gathering nature of their curiosity. When Boulton went searching for
	    the mineral Blue John (to fashion into ormolu-mounted vases), he also collected
	    rocks and fossils in the hope of gauging the age of the earth. And between
	    seeing patients as a physician, Darwin was making notes on everything from
	    telescopic candlesticks to "5-inch worms in cats".  
	    In politics, too, the Lunar Men aspired to change the world. Whigs almost
	    to a man, they supported the American colonists against the government in
	    1776 ("What will become of button-making?" exclaimed Boulton, the society's
	    token Tory, as he contemplated the drop in demand for uniforms); and Wedgwood
	    spoke enthusiastically of 1789 as "that glorious revolution that has taken
	    place in France".  
	    Ironically, this political and religious radicalism indirectly proved to
	    be the Lunar Society's undoing. In 1791, Birmingham's Church-and -King mobs
	    turned their anger on the heretical Priestley (who denied the divinity of
	    Christ), burning his house and forcing him to flee in fear of his life. After
	    that traumatic moment, the society's meetings never quite regained their
	    former energy or conviviality, and the society eventually petered out and
	    died. 
	    Not everything convinces. Uglow stresses that the Lunar Men were mostly
	    non-Oxbridge, non-London, and from "outside the Establishment "- something
	    that proved a real strength, since they were unhampered by old traditions
	    of deference and stuffy institutions". Yet the
	    Hanoverian Establishment was itself highly heterogeneous; and, for their
	    part, the friends were often reliant on aristocratic patronage (the Duke
	    of Bridgewater's, for instance, in their canal projects) and were quite happy
	    to lay obsequiousness on with a trowel whenever it looked advantageous to
	    do so. Wedgwood assiduously toadied to royal patrons and took pride in his
	    title, "Potter to the Queen".  
	    Moreover, while there are moments when the book's torrent of detail threatens
	    to overwhelm the reader, there are also some major omissions. Most seriously,
	    there is not a word about the Deism, the heterodox contemporary religious
	    movement that identified God with Nature. which pervades so many of
	    the Lunar friends' attitudes and beliefs. This remains, however, a magnificently
	    accomplished and enjoyable book. And if it highlights the Lunar Men's prodigious
	    learning, it also demonstrates the exceptional abilities of at least one
	    Lunar Woman. For in the sheer range of Jenny Uglow's erudition, her obvious
	    delight in the inquisitiveness of her savants, and in her high skills as
	    a writer, she has proved herself a worthy member of that distinguished club.
	     
	    John Adamson is a fellow of Peterhouse Cambridge A subject for discussion
	    at the Lunar Society  
	    [The Sunday Telegraph Sep 15 2002] 
	      | 
	 
       
      
       
	
	  DEEP SPACE SECRETS Over 130 molecules have been identified
	    in interstellar space so far, including sugars and ethanol. Now Lewis Snyder
	    and Yi-Jehng Kuan of the National Taiwan University say they have spotted
	    the amino acid glycine. Amino acids in deep space are a particularly important
	    discovery because they link up to form proteins. If the finding stands up
	    to scrutiny it will add oomph to ideas that life exists on other planets,
	    and even that molecules from outer space kick-started life on Earth.  
	     
	    FEEL THE FORCE Pioneer 10 was the first spacecraft to fly past Jupiter. Pioneer
	    11 went on to visit Saturn. Out at the darkest edge of the Solar System beyond
	    Pluto, there should be nothing to slow the probes down except the feeble
	    gravity of the receding Sun - yet a mysterious extra force seems to be tugging
	    on them. "For the life of me, I can't think what it could be," says Michael
	    Martin Nieto from Los Alamos National Laboratory in New Mexico. But "I admit
	    I want it to be something profoundly important, some entirely new physics,"
	    he adds. And he just may be in luck...  
	     | 
	 
       
      
       
       
       
	
	   QUADRATURE OF LUNES 
	    Somewhat younger than Anaxagoras, and coming originally from about the same
	    part of the Greek world, was Hippocrates of Chios. He should not be confused
	    with his still more celebrated contemporary, the physician Hippocrates of
	    Cos. Both Cos and Chios are islands in the Dodecanese group; but Hippocrates
	    of Chios in about 430 B.C. left his native land for Athens in his capacity
	    as a merchant. Aristotle reports that Hippocrates was less shrewd than Thales
	    and that he lost his money in Byzantium through fraud; others say that he
	    was beset by pirates. In any case, the incident was never regretted by the
	    victim, for he counted this his good fortune in that as a consequence he
	    turned to the study of geometry, in which he achieved remarkable success-a
	    story typical of the Heroic Age. Proclus wrote that Hippocrates composed
	    an "Elements of Geometry," anticipating by more than a century the better-known
	    Elements of Euclid. However, the textbook of Hippocrates-as well as another
	    reported to have been written by Leon, a later associate of the Platonic
	    school-has been lost, although it was known to Aristotle. In fact, no
	    mathematical treatise from the fifth century has survived; but we do have
	    a fragment concerning Hippocrates which Simplicius (fi.. ca. 520) claims
	    to have copied literally from the History of Mathematics (now lost) by Eudemus.
	    This brief statement, the nearest thing we have to an original source on
	    the mathematics of the time, describes a portion of the work of Hippocrates
	    dealing with the quadrature of lunes. A lune is a figure bounded by two circular
	    arcs of unequal radii; the problem of the quadrature of lunes undoubtedly
	    arose from that of squaring the circle. The Eudemian fragment attributes
	    to Hippocrates the following theorem: Similar segments of circles are in
	    the same ratio as the squares on their bases. The Eudemian account reports
	    that Hippocrates demonstrated this by first showing that the areas of two
	    circles are to each other as the squares on their diameters. Here Hippocrates
	    adopted the language and concept of proportion which played so large a role
	    in Pythagorean thought. In fact, it is thought by some that Hippocrates became
	    a Pythagorean. The Pythagorean school in Croton had been suppressed (possibly
	    because of its secrecy, perhaps because of its conservative political
	    tendencies), but the scattering of its adherents throughout the Greek world
	    served only to broaden the influence of the school. This influence undoubtedly
	    was felt, directly or indirectly, by Hippocrates. The theorem of Hippocrates
	    on the areas of circles seems to be the earliest precise statement on curvilinear
	    mensuration in the Greek world. Eudemus believed that Hippocrates gave a
	    proof of the theorem, but a rigorous demonstration at that time (say about
	    430 B.C.) would appear to be unlikely. The theory of proportions at that
	    stage probably was established for commensurable magnitudes only. The proof
	    as given in Euclid XII.2 comes from Eudoxus, a man who lived halfway between
	    Hippocrates and Euclid. However, just as much of the material in the first
	    two books of Euclid seems to stem from the Pythagoreans, so it would appear
	    reasonable to assume that the formulations, at least, of much of Books III
	    and IV of the Elements came from the work of Hippocrates. Moreover, if
	    Hippocrates did give a demonstration of this theorem on the areas of circles,
	    he may have been responsible for the introduction into mathematics of the
	    indirect method of proof. That is, the ratio of the areas of two circles
	    is equal to the ratio of the squares on the diameters or it is not. By a
	    reductio ad absurdum from the second of the two possibilities, the proof
	    of the only alternative is established. From this theorem on the areas of
	    circles Hippocrates readily found the first rigorous quadrature of a curvilinear
	    area in the history of mathematics. He began with a semicircle circumscribed
	    about an isosceles right triangle, and on the base (hypotenuse) he constructed
	    a segment similar to the circular segments on the sides of the right triangle.
	    (Fig. 5.1). Because the segments are to each other as squares on their bases,
	    and from the Pythagorean theorem as applied to the right triangle, the sum
	    of the two small circular segments is equal to the larger circular segment.
	    Hence, the difference between the semicircle on AC and the segment ADCE equals
	    triangle ABC. Therefore, the lune ABCD is precisely equal to triangle ABC;
	    and since triangle ABC is equal to the square on half of AC, the quadrature
	    of the lune has been found. 
	     Eudemus describes also an Hippocratean lune quadrature based on an
	    isosceles trapezoid ABCD inscribed in a circle so that the square on the
	    longest side (base) AD is equal to the sum of the squares on the three equal
	    shorter sides AB and BC and CD (Fig. 5.2). Then, if on side AD one constructs
	    a circular segment AEDF similar to those en the three equal sides, lune ABCDE
	    is equal to trapezoid ABCDF. 
	    That we are on relatively firm ground historically in describing the quadrature
	    of lunes by Hippocrates is indicated by the fact that scholars other than
	    Simplicius also refer to this work. Simplicius lived in the sixth century,
	    but he depended not only on Eudemus (fi. ca. 320 B.C.) but also on Alexander
	    of Aphrodisias (fi. ca. A.D. 200), one of the chief commentators on Aristotle.
	    Alexander describes two quadratures other than those given above. (1) If
	    on the hypotenuse and sides of an isosceles right triangle one constructs
	    semicirdes (Fig. 5.3),then the lunes created on the smaller sides together
	    equal the triangle. (2) If on a diameter of a semicircle one constructs an
	    isosceles trapezoid with three equal sides (Fig. 5.4), and if on the three
	    equal sides semicircles are constructed, then the trapezoid is equal in area
	    to the sum of four curvilinear areas: the three equal lunes and a semicircle
	    on one of the equal sides of the trapezoid. From the second of these quadratures
	    it would follow that if the lunes can be squared, the semicircle-hence the
	    circle-can also be squared. This conclusion seems to have encouraged Hippocrates,
	    as well as his contemporaries and early successors, to hope that ultimately
	    the circle would be squared. 
	    
	    
	     
	     
	    CONTINUED PROPORTIONS 
	    The Hippocratean quadratures are significant not so much as attempts at
	    circle-squaring as indications of the level of mathematics at the time. They
	    show that Athenian mathematicians were adept at handling transformations
	    of areas and proportions. In particular, there was evidently no difficulty
	    in converting a rectangle of sides a and b into a square. This required finding
	    the mean proportional or geometric mean between a and b. That is, if a:x
	    = x:b, geometers of the day easily constructed the line x. It was natural,
	    therefore, that geometers should seek to generalize the problem by inserting
	    two means between two given magnitudes a and b. That is, given two line segments
	    a and b, they hoped to construct two other segments x and y such that a:x
	    = x:y = y:b. Hippocrates is said to have recognized that this problem is
	    equivalent to that of duplicating the cube; for if b = 2a, the continued
	    proportions, upon the elimination of y, lead to the conclusion that
	    x3 = 2a3. There are three views on what Hippocrates
	    deduced from his quadrature of lunes. Some have accused him of believing
	    that he could square all lunes, hence also the circle; others think that
	    he knew the limitations of his work, concerned as it was with some types
	    of lunes only. At least one scholar has held that Hippocrates knew he had
	    not squared the circle but tried to deceive his countrymen into thinking
	    that he had succeeded. There are other questions, too, concerning Hippocrates'
	    contributions, for to him has been ascribed, with some uncertainty, the first
	    use of letters in geometric figures. It is interesting to note that whereas
	    he adyanced two of the three famous problems, he seems to have made no progress
	    in the trisecting of the angle, a problem studied somewhat later by Hippias
	    of Elis. 
	    
	     
	     HIPPIAS OF ELIS 
	    Toward the end of the fifth century B.C. there flourished at Athens a group
	    of professional teachers quite unlike the Pythagoreans. Disciples of Pythagoras
	    had been forbidden to accept payment for sharing their knowledge with others.
	    The Sophists, however, openly supported themselves by tutoring fellow
	    citizens-not only in honest intellectual endeavor, but also in the art of
	    "making the worse appear the better." To a certain extent the accusation
	    of shallowness directed against the Sophists was warranted; but this should
	    not conceal the fact that Sophists usually were very widely informed in many
	    fields and that some of them made real contributions to learning. Among these
	    was Hippias, a native of Elis who was active at Athens in the second half
	    of the fifth century B.C. He is one of the earliest mathematicians of whom
	    we have firsthand information, for we learn much about him from Plato's
	    dialogues. We read, for example, that Hippias boasted that he had made more
	    money than any two other Sophists. He is said to have written much, from
	    mathematics to oratory, but none of his work has survived. He had a remarkable
	    memory, he boasted immense learning, and he was skilled in handicrafts. To
	    this Hippias (there are many others in Greece who bore the same name) we
	    apparently owe the introduction into mathematics of the first curve beyond
	    the circle and the straight
	    line.  
	    Proclus and other commentators ascribe to him the curve since known as the
	    trisectrix or quadratrix of Hippias.2 This is drawn as follows: In the square
	    ABCD (Fig. 5.5) let side AB move down uniformly from its present position
	    until it coincides with AC and let this motion take place in exactly the
	    same time that side BA rotates clockwise from its present position until
	    it coincides with DC. If the positions of the two moving lines at any given
	    time are given by A 'B' and BA" respectively and if P is the point of
	    intersection of A 'B' and DA", the locus of P during the motions will be
	    the trisectrix of Hippias-curve APQ in the figure. Given this curve, the
	    trisection of an angle is carried out with ease. For example, if PDC is the
	    angle to be trisected, one simply trisects segments B'C and A 'D at points
	    R, S, T, and U. If lines TR and US cut the trisectrix in V and W respectively,
	    lines VD and WD will, by the property of the trisectrix, divide angle PDC
	    in three equal parts. The curve of Hippias generally is known as the quadratrix,
	    since it can be used to square the circle. Whether or not Hippias himself
	    was aware of this application cannot now be determined. It has been conjectured
	    that Hippias knew of this method of quadrature but that he was unable to
	    justify it. Since the quadrature through Hippias' curve was specifically
	    given later by Dinostratus, we shall describe this work in the next chapter.
	    Hippias lived at least as late as Socrates (~399 B.C.), and from the pen
	    of Plato we have an unflattering account of him as a typical Sophist-vain,
	    boastful, and acquisitive. Socrates is reported to have described Hippias
	    as handsome and learned, but boastful and shallow. Plato's dialogue on Hippias
	    satirizes his show of knowledge, and Xenophon's Memorabilia includes an
	    unflattering account of Hippias as one who regarded himself an expert in
	    everything from history and literature to handicrafts and science. In judging
	    such accounts, however, we must remember that Plato and Xenophon were
	    uncompromisingly opposed to the Sophists in general. It is well to bear in
	    mind also that both Protagoras, the "founding father of the Sophists," and
	    Socrates, the archopponent of the movement, were antagonistic to mathematics
	    and the sciences. With respect to character, Plato contrasts Hippias with
	    Socrates, but one can bring out much the same contrast by comparing Hippias
	    with another contemporary-the Pythagorean mathematician Archytas of Tarentum.
	     
	     1See Bjornbo's article "Hippocrates" in Pauly-Wissowa,
	    Real-Enzykiopadie der klassisehen Altertumswissenschaft, Vol. VIII, p.1796.
	    2An excellent account of this is found in K. Freeman, The Pre-Socratic
	    Philosophers. A Companion to Diels, Fragmente der Vorsokratiker (1949),
	    pp.381-391. See also the article on Hippias in Pauly-Wissowa. op. cit.. Vol.
	    VIII, pp. 1707 ff.  
	    Carl C Boyer " A History of Mathematics " p 65-69   | 
	 
       
      
       
	
	  
	       The Pink "pirate" who sailed the southern stars
	    
	    
	    Comet man Halley also blazed a trail on Earth in a
	    search for Antarctica  | 
	 
	
	  ONE day in l700 a fishing vessel was at
	    work off the Newfoundland coast when a ragged looking craft bore down
	    on it. Fearing piracy, the fisherman opened fire. The result was a torrent
	    of foul language from the "pirate's" commander, the astronomer Edmond Halley.
	     
	    Halley was not only famous for predicting the appearance of his comet. He
	    was also a bold explorer, travelling the world in search of scientific
	    information, who nearly lost his life trying to find Antarctica a
	    century before Captain Cook arrived. 
	    His extraordinary exploits are revealed in the latest issue of Astronomy
	    Now by the scientific historian Dr Ian Seymour, who relates that Halley
	    was not only a precursor of Cook but, in being faced with a mutiny that he
	    himself partly provoked, of Captain Bligh too.  
	    In his second voyage, begun in 1698, he set out in a Royal Naval vessel called
	    the Paramour Pink.
	     "Pinks", flat-bottomed ships 52 feet long and 18 feet broad,
	    -specially designed for sailing in shallow seas and almost unknown in the
	    Navy, were often mistaken for pirates sailing under false colours. 
	    The crew numbered 20, making the vessel extremely cramped, and the first
	    officer, a certain Harrison (no relation to the inventor
	    of the marine chronometer) was a professional seaman who despised the
	    "academic" Halley from whom, he complained with gross unfairness, "much is
	    expected and little or nothing appears". 
	    During this voyage Halley did in fact make extensive observations of the
	    Earth's magnetic fields which, Dr Seymour says, "remained indispensable
	    shipboard companions for more than a century".  
	    But he was an appallingly bad commander. Despite his naval authority, he
	    never had a man flogged and attempted to enforce discipline with sarcastic
	    and foul-mouthed abuse.  
	    Harrison, openly insubordinate, countermanded orders and told the crew that
	    Halley had only been given command because of his wealthy connections, since
	    he was useless for any other occupation. 
	    One day Harrison told Halley in the presence of all the crew that he was
	    "not only uncapable to take charge of a Pink, but even of a longboat". Halley
	    had him confined to his cabin for the rest of the voyage. At the subsequent
	    court martial the Admiralty appeared to recognise Halley's faults, for the
	    mutinous officer escaped with only a reprimand. 
	    On a subsequent voyage in a Pink, this time with a more agreeable first officer,
	    Halley put in at Recife, Brazil, where he fell foul of the English consul,
	    a Mr Hardwicke; whom he afterwards alleged was an imposter. 
	    He told Hardwicke that the purpose of the voyage was to observe the stars
	    in the southern skies. (Halley, in fact, had earlier won election to the
	    Royal Society for identifying 341 southern stars from the murky skies of
	    Saint Helena.)  
	    
	    
	     
	      
		  | 
	       
	      
		
		  Halley: storm-tossed life  | 
	       
	     
	    
	    Hardwicke said this story was too ridiculous to be believed.
	    Citing the suspicious appearance of Halley's ship, he ordered him to be arrested
	    as a pirate. Halley was released after a few hours at the intervention of
	    the city's governor but,incensed and refusing to accept apologies, he set
	    off into stormy seas. 
	    He was soon in the Southern Ocean hoping to find the fabled Lost
	    Continent. The ship's lookouts reported seeing three large islands, unmarked
	    on any map. They were all "flat at the top, covered with snow, milk-white,
	    with perpendicular cliffs all around them". 
	     Surrounded
	    by these icebergs and by thick fog, the ship was soon in deadly peril. For
	    "between 11 and 12 days", Halley wrote in the log, "we were in imminent danger
	    of the inevitable loss of all of us in case we starved, being alone without
	    a consort". They were saved by the smallness of the ship, which made it
	    responsive to controls, and by its shallow draft. 
	    Halley, who lived to 86, was one of the most remarkable scientists of all
	    ages. A friend of Isaac Newton - probably the only friend that cantankerous
	    man ever had - he was influential in securing the publication of his Principia,
	    that basis of all celestial laws. He also discovered the first known globular
	    cluster whose ancient stars today defy our attempts to age the universe.
	     
	    And he was a great character too, as shown by his entertainment, when Astronomer
	    Royal, of the visiting Russian tsar Peter the Great. They ended up drunk
	    in a ditch. 
	     Adrian Berry  
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      The Sun | The
      Solar System |
      The Sky
      at Night | Asimov - "Extraterrestrial Civilisations -
      The Moon |
      Moonshadow's
      Realm  | 
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