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Archimedes Quadrature of the Parabola

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Archimedes Quadrature of the Parabola. Archimedes (287 -212 B.C) ... The quadrature of the parabola using indivisibles and the physical picture of a scale. ... – PowerPoint PPT presentation

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Title: Archimedes Quadrature of the Parabola


1
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.

2
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.

3
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

4
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)

5
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.

6
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.

7
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)

8
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)

9
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)

10
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)
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