Archimedes Quadrature of the Parabola
• Archimedes (287 -212 B.C).
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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:
LATIN:Noli turbare circulos meos.
ENGLISH:Don't disturb my circles.
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 L>K, 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)<(L-K) then K<P which is impossible.
– Assume L<K, 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.
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”
Archimedes
Integrating the parabola with a lever
Given a segment AC of the parabola construct:
1. The middle of the segment D of AC.
2. 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.
3. The triangle AFC which is given by the base AC, the tangent at
C and the line parallel to BD through A.
Then
A. 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 DB=EB and hence NO=NM
B. 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:
MO:OP=CA:AO=CK:KN
C. A(DAFC)=4A(DABC)
Archimedes
Integrating 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 CK=HK.
MO:TG=HK:KN
– 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 2:1.
Archimedes
Integrating 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)=HK:KW=3:1
• 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.
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)
Zeno of Elea (490-425 BC)
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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)
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)
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)