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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.)
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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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