archimedes' quadrature of the parabola - holy...

IntroductionThe problem and Archimedes’ discovery

Some conclusions

Archimedes’ Quadrature of the Parabola

John B. Little

Department of Mathematics and Computer ScienceCollege of the Holy Cross

June 12, 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 Sicily

Studied 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 (discussed by Plutarch, Cicero)



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 (discussed by Plutarch, Cicero)



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 (discussed by Plutarch, Cicero)



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 (discussed by Plutarch, Cicero)



Some conclusions

Figure: Sphere inscribed in cylinder of equal radius

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



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

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

On the Equilibrium of Planes (2 books)On Floating Bodies (2 books)Measurement of a Circle

On Conoids and SpheroidsOn SpiralsOn the Sphere and Cylinder (2 books)



Some conclusions

Surviving works

On the Equilibrium of Planes (2 books)On Floating Bodies (2 books)Measurement of a CircleOn Conoids and Spheroids

On SpiralsOn the Sphere and Cylinder (2 books)



Some conclusions

Surviving works

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

On the Sphere and Cylinder (2 books)



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 Stomachion

Quadrature 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)



Some conclusions


Book of Lemmas, The Sand-Reckoner, TheCattle-Problem, the StomachionQuadrature of the Parabola

The 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)



Some conclusions


Book of Lemmas, The Sand-Reckoner, TheCattle-Problem, the StomachionQuadrature of the ParabolaThe Method of Mechanical Theorems

The 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)



Some conclusions


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 otherworks

1906 – a palimpsest prayerbook created about 1229 CE (areused manuscript) was found by to contain substantialportions (a 10th century CE copy from older sources)



Some conclusions


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)



Some conclusions

A page from the palimpsest



Some conclusions

Scientific context

Archimedes flourished at height of Hellenistic period;Greek kingdoms (fragments of Alexander’s empire) inEgypt, Syria, Syracuse strong patrons of science andmathematics

State 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



Some conclusions

Scientific context

Archimedes flourished at height of Hellenistic period;Greek kingdoms (fragments of Alexander’s empire) inEgypt, Syria, Syracuse strong patrons of science andmathematicsState 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



Some conclusions

Scientific context

Archimedes flourished at height of Hellenistic period;Greek kingdoms (fragments of Alexander’s empire) inEgypt, Syria, Syracuse strong patrons of science andmathematicsState 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 colleagues

Example: Letter to Eratosthenes (in Alexandria) at start ofThe Method



Some conclusions

Scientific context

Archimedes flourished at height of Hellenistic period;Greek kingdoms (fragments of Alexander’s empire) inEgypt, Syria, Syracuse strong patrons of science andmathematicsState 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



Some conclusions

Goals for this talk

Discuss a problem treated by Archimedes in The Methodand (later?) in Quadrature of the Parabola and the twodifferent solutions Archimedes gave

Think about the relation of the methods Archimedesdeveloped and later innovations in mathematics such aslimits, integral calculus, etc.Get a glimpse of the way Archimedes thought about whathe was doing



Some conclusions

Goals for this talk

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, integral calculus, etc.

Get a glimpse of the way Archimedes thought about whathe was doing



Some conclusions

Goals for this talk

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, integral calculus, etc.Get a glimpse of the way Archimedes thought about whathe was doing



Some conclusions

What is the area of a parabolic segment?

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

3area(∆ABC)



Some conclusions


Let C have same x-coordinate as the midpoint of AB

The area of the parabolic segment = 43area(∆ABC)



Some conclusions


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

3area(∆ABC)



Some conclusions

“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 (see 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!



Some conclusions


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 (see Book 12 ofElements), based on work of Eudoxus

A 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!



Some conclusions


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 (see 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!



Some conclusions


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 (see 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 Method

We will look at both of these using modern notation – bothquite interesting!



Some conclusions


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 (see 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!



Some conclusions

What’s special about this C?

Can show easily with calculus (and less so using thegeometric definition of a parabola): The tangent to theparabola at C is parallel to AB

So 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



Some conclusions


Can show easily with calculus (and less so using thegeometric definition of a parabola): The tangent to theparabola at C is parallel to ABSo for this C, and this C only, the heights of the triangleand the parabolic segment are equal

The point C is called the vertex of the segment



Some conclusions


Can show easily with calculus (and less so using thegeometric definition of a parabola): The tangent to theparabola at C 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



Some conclusions

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 + · · · + 1

4n A0

Moreover, 43A0 − Sn = 1

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

3A0



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 + · · · + 1

4n 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, so

Archimedes 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

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 TheMethod

We’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


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 at

Key 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


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 intext (a Maple issue!)CF 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): Thecentroid of ∆AFC islocated at a point W alongCK with CK = 3 · KW .



Some conclusions

“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 to Eratosthenes

“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



Some conclusions

What does this tell us?

Can certainly say that Archimedes anticipated many ideasof integral calculus in this work

But 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



Some conclusions


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 segments

So he also worked out the first “method of exhaustion”proof to supply an argument that would satisfy themathematicians of his day



Some conclusions


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



Some conclusions

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

Archimedes’ place in mathematical history

It should be clear from these examples that Archimedeswas one of the greatest mathematicians in all of our knownhistory – his work shows extremely strong originality,masterful use of an intuitive physical analogy, and technicalpower

But he was so far ahead of his time that within 100 years orso of his death very few, if any, people were reading himwith complete understandingAlmost miraculous that this has survivedAlso extremely poignant to think what might have been ifothers had been better prepared to follow his lead at thetime!



Some conclusions


It should be clear from these examples that Archimedeswas one of the greatest mathematicians in all of our knownhistory – his work shows extremely strong originality,masterful use of an intuitive physical analogy, and technicalpowerBut he was so far ahead of his time that within 100 years orso of his death very few, if any, people were reading himwith complete understanding

Almost miraculous that this has survivedAlso extremely poignant to think what might have been ifothers had been better prepared to follow his lead at thetime!



Some conclusions


It should be clear from these examples that Archimedeswas one of the greatest mathematicians in all of our knownhistory – his work shows extremely strong originality,masterful use of an intuitive physical analogy, and technicalpowerBut he was so far ahead of his time that within 100 years orso of his death very few, if any, people were reading himwith complete understandingAlmost miraculous that this has survived

Also extremely poignant to think what might have been ifothers had been better prepared to follow his lead at thetime!



Some conclusions


It should be clear from these examples that Archimedeswas one of the greatest mathematicians in all of our knownhistory – his work shows extremely strong originality,masterful use of an intuitive physical analogy, and technicalpowerBut he was so far ahead of his time that within 100 years orso of his death very few, if any, people were reading himwith complete understandingAlmost miraculous that this has survivedAlso extremely poignant to think what might have been ifothers had been better prepared to follow his lead at thetime!


archimedes' quadrature of the parabola - holy...

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