Archimedes Quadrature of the Parabola

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Archimedes Quadrature of the Parabola

• Archimedes (287 -212 B.C).
• Lived in Syracuse on Sicily.
• Invented many machines which were used as engines
  of war
• Invented the compound pulley and the water snail.
• Archimedes' principle weight of a body immersed
  in a liquid is equal to the displaced liquid.
  (Eureka!)
• In Measurement of the Circle Archimedes shows
  that the exact value of p lies between the values
  310/71 and 31/7. This he obtained by
  circumscribing and inscribing a circle with
  regular polygons having 96 sides
• Discovered the lever principle Give me a place
  to stand on and I will move the earth.
• He was killed in 212 BC during the capture of
  Syracuse by the Romans in the Second Punic War.
• Legendary last words of Archimedes
• GREEK
• LATINNoli turbare circulos meos.
• ENGLISHDon't disturb my circles.

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Archimedes Quadrature of the Parabola

• Proposition 20 Gives the basic construction for
  the iterative process.
• Proposition 22 Shows that the area of the
  segment is an upper bound for the area of the
  inscribed polygon.
• Proposition 23 Gives the difference between the
  expected area of the segment and the area of a
  polygon.
• Proposition 24 Is the main statement that the
  area of the segment is given by 4/3 of the area
  of the triangle over the segment with the
  greatest height.
• To show 24 Let A be the area of the triangle K
  4/3 A and L be the area of the segment.
• Assume LgtK, then this is not possible, since we
  can make the difference between L and its
  approximations by polygons if area P arbitrarily
  small. If (L-P)lt(L-K) then KltP which is
  impossible.
• Assume LltK, then this is not possible either,
  since the approximation by polygons can be
  arbitrarily close to K by Prop. 23, thus will
  eventually be bigger than L, but by Prop.22 L is
  an upper bound for these approximations.
• The method is a double reduction ad absurdum. It
  avoids taking infinite limits. It assumes
  properties of real numbers though.

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Archimedes The Method

• The Method contains Archimedes intuition about
  mathematical theorems based on physics and
  indivisibles.
• It was lost until J.L. Heiberg found it in a
  monastery in Jerusalem in 1899
• The quadrature of the parabola using indivisibles
  and the physical picture of a scale.
• Archimedes gave the laws for the lever in On
  balances and levers.
• Two magnitudes balance at distances
  reciprocally proportional to the magnitudes
• Give me a place to stand on and a lever long
  enough and I will move the world

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ArchimedesIntegrating the parabola with a lever

• Given a segment AC of the parabola construct
• The middle of the segment D of AC.
• The triangle ABC where B is the intersection of
  the parabola with the line through D which is
  parallel to the axis, i.e. the diameter.
• The triangle AFC which is given by the base AC,
  the tangent at C and the line parallel to BD
  through A.
• Then
• The line CB cuts all parallel segments MO from
  the side CF to the base AC to BD in their middle
  point N. Since CE is tangent and thus DBEB and
  hence NONM
• For any segment MO as above let P be the
  intersection point with the parabola and K the
  middle point of CF by properties of the
  parabola
• MOOPCAAOCKKN
• A(DAFC)4A(DABC)

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ArchimedesIntegrating the parabola with a lever

• Extend CK to double its length CH and use CH as
  a lever fixed at K.
• weigh the segment OP by putting its center of
  gravity at H and let it hang straight down i.e.
  parallel to DB and call the resulting segment TG.
  Then it is balanced with the segment MO, since
  CKHK.
• MOTGHKKN
• In this way hang all the segments
  (indivisibles) making up the segment of the
  parabola at K. They will be in equilibrium with
  all the segments (indivisibles) of the triangle
  ACF left at their place or equivalently with the
  triangle affixed to its center of mass W, which
  is the point on KC at 1/3 of the total distance
  KC from K.
• Recall that the center of mass of a triangle is
  at the intersection of its medians and divides
  the medians 21.

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ArchimedesIntegrating the parabola with a lever

• In balance, we obtain that the weight which we
  think of as area of DACF equals 3 the weight of
  the segment.
• A(DACF)A(segment ABC)HKKW31
• Therefore
• A(segment ABC)4/3 A(DACB)
• Archimedes comments that since he used the
  indivisibles, this is a good reason to believe in
  the result, but it is not a proof.

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Zeno of Elea (490-425 BC)

• Zeno of Elea (490-425 BC) was a student of
  Parmenides (515-450 BC).
• He is famous for his paradoxes on motion and the
  infinite. He had a book of 40 paradoxes.
• Aristotle features 4 of his paradoxes on motion
  in Physics. The Dichotomy-, The Achilles-, The
  Arrow-, and The Stadium Paradox-.
• The Dichotomy
• The first asserts the non-existence of motion on
  the ground that that which is in locomotion must
  arrive at the half-way stage before it arrives at
  the goal. (Aristotle Physics, 239b11)

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Zeno of Elea (490-425 BC)

• The Archilles
• the slower when running will never be overtaken
  by the quicker for that which is pursuing must
  first reach the point from which that which is
  fleeing started, so that the slower must
  necessarily always be some distance ahead.
  (Aristotle, Physics 239b23)
• The second argument was called "Achilles,"
  accordingly, from the fact that Achilles was
  taken as a character in it, and the argument
  says that it is impossible for him to overtake
  the tortoise when pursuing it. For in fact it is
  necessary that what is to overtake something,
  before overtaking it, first reach the limit
  from which what is fleeing set forth. In the
  time in which what is pursuing arrives at this,
  what is fleeing will advance a certain interval,
  even if it is less than that which what is
  pursuing advanced  . And in the time again in
  which what is pursuing will traverse this
  interval which what is fleeing advanced, in
  this time again what is fleeing will traverse
  some amount  . And thus in every time in which
  what is pursuing will traverse the interval
  which what is fleeing, being slower, has already
  advanced, what is fleeing will also advance some
  amount. (Simplicius(b) On Aristotle's Physics,
  1014.10)

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Zeno of Elea (490-425 BC)

• The Arrow
• The third is that the flying arrow is at rest,
  which result follows from the assumption that
  time is composed of moments  . he says that if
  everything when it occupies an equal space is at
  rest, and if that which is in locomotion is
  always in a now, the flying arrow is therefore
  motionless. (Aristotle Physics, 239b.30) Zeno
  abolishes motion, saying "What is in motion moves
  neither in the place it is nor in one in which it
  is not". (Diogenes Laertius Lives of Famous
  Philosophers, ix.72)

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Zeno of Elea (490-425 BC)

• The Arrow
• The third is that the flying arrow is at rest,
  which result follows from the assumption that
  time is composed of moments  . he says that if
  everything when it occupies an equal space is at
  rest, and if that which is in locomotion is
  always in a now, the flying arrow is therefore
  motionless. (Aristotle Physics, 239b.30)
• The Stadium
• The fourth argument is that concerning equal
  bodies AA which move alongside equal bodies in
  the stadium from opposite directions -- the ones
  from the end of the stadium CC, the others from
  the middle BB -- at equal speeds, in which he
  thinks it follows that half the time is equal to
  its double. And it follows that the C has passed
  all the As and the B half so that the time is
  half  . And at the same time it follows that the
  first B has passed all the Cs. (Aristotle
  Physics, 239b33)