3 Construction Problems of Antiquity and the Quadratrix

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3 Construction Problems of Antiquity and the
Quadratrix

• MATH 4/5620 FA04
• Chapter 3, Section 4 5

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Second half of 5th Century

• Hippocrates of Chios, a mathematician, dominated
  during this time period (460-380 B.C.).
• Began his life as a merchant and ended it as a
  teacher.
• Robbed of his money and went to Athens to courts.
• While there, he attended various lectures.

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Hippocrates

• Had no Pythagorean teacher in that he was not
  part of the Pythagorean school.
• He attained a proficiency in geometry and was one
  of the first to support himself openly by
  accepting fees for teaching mathematics.
• If he had been taught by Pythagoreans, then he
  betrayed their trust.

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Mathematics movement

• By mid 5th century, so many geometric theorems
  had been established that it became necessary to
  tighten the proofs and put things in logical
  order.
• It was noted that Hippocrates composed a work on
  the elements of geometry anticipating the
  Elements by more than a century but no record
  of it exists.

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Contributions of Hippocrates

• He originated the pattern of presenting geometry
  as a chain of propositions, a form in which other
  propositions can be derived on the basis of
  earlier ones.
• He also introduced the use of letters of the
  alphabet to designate points and lines in
  geometric figures.
• His fame rests on one of the 3 problems of
  Antiquity.

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3 problems of antiquity

• The quadrature of the circle sometimes called
  the squaring of the circle
• The duplication of the cube
• The trisection of a general angle

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The 3 problems

• Hippocrates main work was on the first problem
  -- the squaring of the circle. Is it possible
  to construct a square whose area shall be equal
  to the area of a given circle?
• PLATO (429-348 B.C.) indicated only a compass and
  straightedge could be used to complete the
  construction.

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Squaring Circle Problem

• In the 19th century, mathematicians proved that
  it is impossible to square the circle by compass
  and straightedge alone.
• With such limitations, it turned out that the
  problem uses algebra rather than geometry and
  also involved concepts unknown during the time
  period.

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• It involved constructing a line segment whose
  length is
• times the radius of the circle.
• It needed to be shown that square root of pi was
  not a constructible length this argument hinges
  on the transcendental nature of pi (pi is not
  the root of any polynomial equation with rational
  coefficients). proved in late 1800s

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Quadratrix

• Hippias of Elis (425 B.C.) invented the
  quadratrix (a sliding apparatus) for the purpose
  of squaring the circle. His solution was
  legitimate, but did not satisfy Platos
  restrictions.
• Hippocrates discovered that there are certain
  plane regions with curved boundaries (LUNES) that
  are squarable.

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• Hippocrates showed that two lunes (moon-shaped
  figure bounded by two circular arcs of unequal
  radii) could be drawn, whose areas were together
  equal to the area of a right triangle (text, p.
  117).
• Having shown the lune could be squared,
  Hippocrates tried to square the circle by a
  similar argument (use of isos. Trapezoid) text,
  p. 118.

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• It is believed that Hippocrates had thought he
  solved the problem, but there was a mistake in
  assuming that every lune can be squared (only
  possible in the case he presented).
• Questionable whether he really thought he had
  solved it since he was such a good
  mathematician.

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2nd Problem

• Duplication of the cube how it originated is
  unclear
• Possible that the Pythagoreans extended the
  doubling of the square (upon the diagonal of a
  square, a new square is constructed where it has
  exactly twice the area of the original square)
  problem into 3 dimensions.

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More common tale

• The Athenians appealed to the oracle at Delos in
  430 B.C. to learn what they should do to
  alleviate a devastating plague. The reply was to
  double the size of the altar (shape of cube) of
  Apollo.
• Builders constructed a cube whose edge was twice
  as long as the edge of the altar.
• Legend has it that the plague was made worse.

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Duplication of cube (cont.)

• Consultation with Plato he responded that The
  god has given this oracle, not because he wanted
  an altar of double the size, but because he
  wished in setting this task before them to
  reproach the Greeks for their neglect of
  mathematics and their contempt of geometry.
• Often called the Delian problem.

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• First real progress was made by Hippocrates he
  showed that it can be reduced to finding, between
  a given line and another line twice as long, two
  mean proportionals.
• Present notation for this text, p. 120
  (geometric proportions)

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Trisection of an angle

• Easy enough to bisect an angle so the thought was
  trisection would fall in place.
• Hippocrates did NOT work on this problem.
• Some angles can be trisected for example the
  case of the right angle.
• In the early 1800s, the first rigorous proof
  emerged that it was impossible to trisect any
  angle by compass and straightedge.

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Quadratrix of Hippias

• Invented to trisect an angle
• Hippias was one of the first to teach for money
  (like Hippocrates)
• Most who taught for money (sophists) acquired
  their learning and expertise from travels.
• Wealthy took pride in hiring the sophists.

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Quadratrix of Hippias

• It is the first example of a curve that could not
  be drawn by compass and straightedge
• Had to be plotted point by point (text, p. 126).
• The quadratrix can be used to trisect an angle.
• Was actually used to find a square equal in area
  to a given circle (more complex).

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Academy of Plato

• Sophists give up practices to establish
  themselves in Athens.
• New schools open main one was the Academy of
  Plato (where Aristotle studied).
• It was through Plato that mathematics reached the
  place in higher education that it holds today
  mathematics furnished the finest training of the
  mind.
• The Museum (Ptolemy in Alexandria) rivaled the
  Academy

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Problem Trisecting an angle

• Tomahawk method (text, p. 131)
• Limacon method (text, p. 132)