archimedes and the quadrature of the parabola -...

IntroductionThe problem and Archimedes’ discovery

Some conclusions

Archimedes and the Quadrature of theParabola

John B. Little

MATH 110 – Topics in MathematicsMathematics Through Time

November 1, 2013

John B. Little Archimedes’ Quadrature of the Parabola


Some conclusions

Outline

1 Introduction

2 The problem and Archimedes’ discovery

3 Some conclusions



Some conclusions

Who was Archimedes?

Lived ca. 287 - 212 BCE, mostly in Greek city of Syracusein SicilyStudied many topics in what we would call mathematics,physics, engineering (less distinction between them at thetime)We don’t know much about his actual life; much of his laterreputation was based on somewhat dubious anecdotes,e.g. the “eureka moment,” inventions he was said to haveproduced to aid in defense of Syracuse during Romansiege in which he was killed, etc.Perhaps most telling: we do know he designed atombstone for himself illustrating the discovery he wantedmost to be remembered for (still existed in later Romantimes – discussed by Plutarch, Cicero)



Some conclusions

Figure: Sphere inscribed in cylinder of equal radius

3Vsphere = 2Vcyl and Asphere = Acyl (lateral area)



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Recognized in antiquity as a master

“In weightiness of matter and elegance of style, no classicalmathematics treatise surpasses the works of Archimedes. Thiswas recognized in antiquity; thus Plutarch says of Archimedes’works:‘It is not possible to find in all geometry more difficult andintricate questions, or more simple and lucid explanations.Some ascribe this to his genius; while others think thatincredible effort and toil produced these, to all appearances,easy and unlaboured results.’ ”

A. Aaboe, Episodes from the early history of mathematics



Some conclusions

Surviving works

On the Equilibrium of Planes (2 books)On Floating Bodies (2 books)Measurement of a CircleOn Conoids and SpheroidsOn SpiralsOn the Sphere and Cylinder (2 books)



Some conclusions

Surviving works, cont.

Book of Lemmas, The Sand-Reckoner, TheCattle-Problem, the StomachionQuadrature of the ParabolaThe Method of Mechanical TheoremsThe Method was long known only from references in otherworks1906 – a palimpsest prayerbook created about 1229 CE (areused manuscript) was found by to contain substantialportions (a 10th century CE copy from older sources)



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A page from the palimpsest



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Scientific context

Archimedes flourished at height of Hellenistic period;Greek kingdoms (fragments of Alexander’s empire) inEgypt, Syria strong patrons of science and mathematics,as was SyracuseState of the art in mathematics: 2nd or 3rd generation afterEuclid, rough contemporary of Apollonius (best known forwork on conic sections)Part of an active scientific community in fairly closecommunication; for instance several of the works citedabove start with letters to colleaguesExample: Letter to Eratosthenes (in Alexandria) at start ofThe Method (will discuss this later)



Some conclusions

Goals for this class

Discuss a problem treated by Archimedes in The Methodand (later?) in Quadrature of the Parabola and the twodifferent solutions Archimedes gaveThink about the relation of the methods Archimedesdeveloped and later innovations in mathematics such aslimits, integrals, etc. some of you have studied if you havetaken calculusEven if you have not seen some of this before, main goal isto appreciate how far Greek mathematics advanced after,and on the basis of Euclid’s Elements



Some conclusions

Conic Sections

the curves obtained as plane slices of a cone:circles, ellipses, parabolas, hyperbolas (named byApollonius of Perga, ca. 200 BCE)Euclid did not discuss these in the ElementsBut he wrote another text on Conics in 4 books, now lost.We know a fair amount about what it must have containedfrom references in other works (e.g. the texts ofArchimedes that are the basis of what we will discusstoday).



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Conic Sections



Some conclusions

Archimedes’ problem: What is the area of aparabolic segment?

Let C have same x-coordinate as the midpoint of ABThe area of the parabolic segment = 4

3area(∆ABC)



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“Quadrature of the Parabola”

In the work in the title of this slide, Archimedes gives twodifferent arguments for this statement,One using the “method of exhaustion” – part of standardrepertoire of Euclidean mathematics (laid out in Book 12 ofElements), based on work of EudoxusA recurring theme in his mathematical work (used it as astandard method to obtain areas and volumes,approximate the value of π, etc.)One by the novel method also described in The MethodWe will look at both of these using modern notation – bothquite interesting!



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What’s special about this C?

Can show in several ways: The tangent to the parabola atC is parallel to ABSo for this C, and this C only, the heights of the triangleand the parabolic segment are equalThe point C is called the vertex of the segment



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Sketch of Archimedes’ first proof

Now have two smaller parabolic segments; let their verticesbe C1 and C2 and construct triangles ∆AC1C and ∆CC2B



Some conclusions

The first proof, continued

Archimedes shows that each of those triangles has area 18

the area of ∆ABC, so

area(∆AC1C) + area(∆CC2B) =14

area(∆ABC).

We do this construction some finite number n of times andadd together the areas of all the triangles,



Some conclusions

First proof, continued

letting A0 be the original area, A1 = 14A0 the area of the

first two smaller triangles, then A2 = 14A1 = 1

42 A0 the areaof the next four, etc.we have total area

Sn = A0 +14

A0 +142 A0 + · · · +

14n A0

Moreover, 43A0 − Sn = 1

3An, so as n increases withoutbound, Sn tends to 4

3A0



Some conclusions

First proof, concluded

On the other hand, as n increases without bound, thetriangles constructed above fill out all of the area in theparabolic segment, soArchimedes concludes that the area of the parabolicsegment = 4

3A0 = 43area(∆ABC) as claimed before.

We would derive all this, of course, using the usualproperties of finite and infinite geometric series:

area of segment =∞∑

n=0

A0

4n =A0

1 − 14

=4A0

3.

Archimedes had to do it all in an ad hoc way because thatgeneral theory did not exist yet!



Some conclusions

Archimedes’ second proof

Now, the remarkable second proof that Archimedespresents both in Quadrature of the Parabola and in TheMethodWe’ll follow the presentation in The Method (in Heath’stranslation) – this seems (to me) to be a less polished formof the argument and even more instructive about whatArchimedes was getting atKey Idea: Apply a physical analogy – will compute the areaof the parabolic segment essentially by considering it as athin plate, or “lamina,” of constant density and “weighing iton a balance” in an extremely clever way



Some conclusions

The construction

Note: ∆ in figure = D intextCF is tangent to theparabola at CB is the vertexO is an arbitrary pointalong the line from A to CED, MO, FA are parallel tothe axis of the parabola



Some conclusions

First observation

By construction of thevertex B, area(∆ADB) =area(∆CDB)

“Well-known” property ofparabolas (at least inArchimedes’ time!):BE = BD, NM = NO,KA = KF∴ area(∆EDC) =area(∆ABC) and bysimilar triangles,

area(∆AFC) = 4·area(ABC)



Some conclusions

Next steps

Produce CK to H makingCK = KHConsider CH “as the bar ofa balance” with K atfulcrumKnown facts abouttriangles (proved byArchimedes earlier): Thecenter of mass (“balancepoint”) of ∆AFC is locatedat a point W along CK withCK = 3 · KW .



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“Dissection” step

Consider PO (think of athin strip approximating thepart of the parabolicsegment with somethickness ∆x)Properties of parabolasand similar triangles implyMO : PO = AC : AO =CK : KN = HK : KNPlace segment TG = POwith midpoint at H.Then Archimedes’ law ofthe lever (fulcrum at K )says TG balances MO



Some conclusions

Conclusion of the proof

Do this for all “strips” likePO. We get that ∆ACFexactly balances the wholecollection of vertical stripsmaking up the parabolicsegment.By the law of the leveragain, area(∆ACF ) :area of segment = HK :KW = 3 : 1Since area(∆ACF ) =4 · area(∆ABC), the proofis complete.



Some conclusions

From the introductory letter of The Method toEratosthenes

“Seeing moreover in you, as I say, an earnest student, a man ofconsiderable eminence in philosophy, and an admirer ofmathematical inquiry, I thought fit to write out for you andexplain in detail ... a certain method by which it will be possiblefor you to get a start to enable you to investigate some of theproblems in mathematics by means of mechanics. Thisprocedure is, I am persuaded, no less useful even for the proofof the theorems themselves; for certain things first becameclear to me by a mechanical method, although they had to bedemonstrated by geometry afterwards because theirinvestigation by the said method did not furnish an actualdemonstration.”Translation by T.L. Heath



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What does this tell us?

Can certainly say that Archimedes anticipated many ideasof integral calculus in this workBut he was also careful and realized that he didn’t have acomplete justification for the idea of “balancing” an areawith a collection of line segmentsSo he also worked out the first “method of exhaustion”proof to supply an argument that would satisfy themathematicians of his day



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Archimedes’ place in mathematical history

Most historians of mathematics agree Archimedes was oneof the greatest mathematicians in all of our known history –his work shows extremely strong originality, masterful useof an intuitive physical analogy, and technical powerBut he was so far ahead of his time that eventually veryfew, if any, people were reading him with completeunderstandingAlmost miraculous that his works have survivedAlso extremely poignant to think what might have been ifothers had been better prepared to follow his lead at thetime!


archimedes and the quadrature of the parabola -...

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