Archimedes Quadrature of the Parabola Archimedes 287 212

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: 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)=4 A(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 which is in locomotion must arrive at the half-way stage before it arrives at the goal. (Aristotle Physics, 239 b 11)

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 239 b 23) 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, 239 b. 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, 239 b. 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, 239 b 33)