archimedes of syracuse - university of florida · works of archimedes conclusion archimedes of...

The EraWorks of Archimedes

Conclusion

Archimedes of Syracuse

Douglas Pfeffer

Douglas Pfeffer Archimedes of Syracuse


Conclusion

Table of contents

1 The Era

2 Works of Archimedes

3 Conclusion



Conclusion

Outline

1 The Era


3 Conclusion



Conclusion

The Punic Wars



Conclusion

The Punic Wars

The Punic1 Wars were a series of wars between Carthage andRome, occurring between 264 BCE and 146 BCE

The First Punic War: 264 BCE - 241 BCEThe Second Punic War: 218 BCE - 201 BCEThe Third Punic War: 149 BCE - 146 BCE

By 146 BCE, the Romans had completely defeated theCarthaginians

1The term Punic comes from the Latin word Punicus, meaning‘Carthaginian’ – referencing the Carthaginians’ Phoenician ancestry.



Conclusion

Siege of Syracuse

In 214 BCE, in the middle of the Second Punic War, Romeattempted to capture the Kingdom of Syracuse

The Roman forces, led by General Marcellus, laid siege to theisland for three yearsSyracuse, known for its fortifications and great walls, wasalmost impenetrable

Defending his city, Archimedes did all he could to fend off theRoman invaders

In particular, he used his knowledge of pulleys and levers toinvent war machines like catapults and devices that could setfire to Roman ships from afar



Conclusion

Siege of Syracuse

Thomas Ralph Spence. Archimedes directing the defenses of Syracuse. 1895.



Conclusion

Siege of Syracuse

In 212 BCE, the Romans managed to invade the city

Despite direct orders from Marcellus to spare Archimedes, hewas slain by a Roman soldierArchimedes was reportedly 75 at the time of his death, andthus was born in 287 BCE



Conclusion

Siege of Syracuse

Thomas Degeorge. Death of Archimedes. 1815.



Conclusion


Domenico Fetti. Archimedes Thoughtful. 1620.



Conclusion


Respect for Archimedes’ talent and accomplishments persistedthroughout the centuries after his death.

In the 18th century, the philosopher Voltaire would write:

“There was more imagination in the head ofArchimedes than in that of Homer.”

We now take a look at some of his work and discuss hiscontributions to mathematics...



Conclusion

Outline

1 The Era


3 Conclusion



Conclusion

On the Equilibriums of Planes

The Law of the Lever:Archimedes further developed what is now called theArchimedean axiom of symmetry:

Bilaterally symmetric bodies are in equilibrium

Generalizing this static principle, Archimedes developed ageneral understanding of levers

About a century earlier, Aristotle had published Physics, buthis discussion of the lever was developed via speculation andnon-mathematical reasoning

Archimedes developed it much like Euclid’s Elements, viapostulates and deductive reasoning.



Conclusion

On the Equilibriums of Planes

Another physics treatise, On Floating Bodies, was a two-bookwork on fluids and buoyancy.

In it, he noted the following: “Any object, wholly or partiallyimmersed in a fluid, is buoyed up by a force equal to theweight of the fluid displaced by the object.”In De Architectura, a book written by the Roman architectMarcus Vitruvius Pollo some time during the first centuryBCE, a fun story involving Archimedes was told about how hisintellect ousted a fraudulent blacksmith

Much of Archimedes’ work in these two works laid thefoundation for the mathematical underpinning of physics

For his contributions to physics and its mathematicaljustification, he is sometimes referred to as the Father ofMathematical Physics.



Conclusion

Sand-Reckoner

Recall that the ancient Greek mathematicians drew a sharpdistinction between logistic and arithmetic

In particular, logistic was somehow looked down upon

In the mid-third century BCE, Aristarchus of Samos proposeda heliocentric model of the universe

Archimedes would later note, “His [Aristarchus’] hypothesesare that the fixed stars and the Sun remain unmoved, that theEarth revolves about the Sun on the circumference of a circle,the Sun lying in the middle of the orbit, and that the sphere offixed stars, situated about the same center as the Sun, is sogreat that the circle in which he supposes the Earth to revolvebears such a proportion to the distance of the fixed stars as thecenter of the sphere bears to its surface.”



Conclusion

Sand-Reckoner

In his work Psammites (or, Sand-Reckoner), Archimedesclaimed he could write down a number greater than thenumber of grains of sand required to fill Aristarchus’ universe

His computation led to an estimate of 1063 grains

Recall that it was around the third century BCE that Greeknumeration switched to the Ionian system with 27 distinctletters

With no positional notation, this system only counted up to amyriad

M = ‘myriad ′ = 10, 000 = 104

Archimedes’ work led to him extending this numerationsystem in an incredible way

A clear contribution to logistic



Conclusion

Sand-Reckoner

Archimedes extended naturally to a myriad-myriad, essentiallya word for the value 108.

Working in this ‘base’, he defined orders in the following way:For a given number x , it was in order n if:

108(n−1) ≤ x < 108n

where x is order 1 if x < 108.

He did this up until a myriad-myriad of orders: up to 108·108

Then he defined values less than 108·108to be of period one.

Continuing, he defined values up to a myriad-myriad of periodsThe largest value he wrote down was essentially(

(108)(108))(108)

= 108·1016

.



Conclusion

Measurement of the Circle

In this treatise, Archimedes provided an estimate for π.

He observed the following:

Start with a regular n-gon inscribed in a circle (he used ahexagon) and let Pn denote its perimeter.Circumscribe the circle with a regular n-gon and let pn denoteits perimeter.Now, do the same for a 2n-gon:



Conclusion


It turns out that P2n and p2n can be found recursively:

P2n =2pnPn

pn + Pnand p2n =

√pnP2n.

His estimates for square roots utilized the Babylonianalgorithm



Conclusion


Similar recursive formulae exist for the areas of each polygon.

Denoted An and an respectively for the area of the inner n-gonand outer n-gon

Starting with the hexagon, Archimedes carefully computedeach iteration until he had approximated P96, p96,A96 and a96

From here he provided bounds for π:

3 1071 < π < 3 10

70

This estimate was better than both the Egyptians and theBabylonians



Conclusion

On Spirals

Archimedes was also enchanted by the problems of Greekantiquity

In an effort to trisect the angle, he investigated theArchimidean-spiral

In modern, polar terms: r = aθ

Of note: Archimides himself attributed the spiral to a friend ofhis, Conon of Alexandria



Conclusion

On Spirals

To trisect an angle:

Interestingly, the spiral can also be used to square the circle.



Conclusion

Quadrature of the Parabola

Conic sections had been known for almost a century byArchimedes’ time, yet no one had attempted to find theirareas

In Proposition 17 of Quadrature of the Parabola, he provideda construction to compute the area of any segment of aparabola:

Start by taking an arbitrary parabola:



Conclusion

Quadrature of the Parabola

In the preamble of this treatise, we find the Axiom ofArchimedes:

“That the excess by which the greater of two unequal areasexceeds the less can, by being added to itself, be made toexceed any given finite area.”This axiom, sometimes referred to as the archimedean propertyof R, is an absolute staple

For example, it is found in part (a) of Theorem 1.20 inRudin’s Principles of Mathematical Analysis



Conclusion

On Conoids and Spheroids

Among other things, this treatise contains a proof that thearea of the ellipse given by

x2

a2 + y2

b2 = 1

is equal to πab.

Unlike the parabola, where he computed the area of anysegment of the curve, Archimedes was unable to do the samefor the ellipse and hyperbola

Through the modern lens, we see that the integral necessary tocompute the area of parabolic segments involves onlypolynomialsThe respective integrals for ellipses and hyperbolas, however,involve transcendental functions.



Conclusion

On the Sphere and Cylinder

Archimedes himself seemed most enchanted by this particulartreatise

In fact, he asked the following to be carved on his tombstone:



Conclusion


This is due to the following discovery:

Given,

It follows that,

Vcylinder

Vsphere=

SAcylinder

SAsphere=

3

2.

Archimedes claimed that this fact was unknown before him.



Conclusion


In Book II of this treatise, Archimedes gave severalcomputations on how to cut a sphere such that:

The two parts’ surface areas were in a desired ratio with oneanotherThe two parts’ volumes were in a desired ratio with oneanother

This question was much more difficult.

It is theorized that these constructions were inspired byEuclid’s Division of Figures



Conclusion

Book of Lemmas

In this treatise, he tackled ‘basic’ problems that were notcategorized as ‘higher’ math

For example, the quadrature of arbelos of the shoemaker’sknife:



Conclusion

Book of Lemmas

Archimedes showed that the area of the arbelos is equal tothat of the circle PRCS

Another example given in his Book of Lemmas is another‘construction’ for trisecting an angle.



Conclusion

The Method

Recall that Euclid’s Elements survived in many Greek andArabic manuscripts,

Archimedes’ treatises persist via a single Greek original fromthe 1700s, copied from a 9th or 10th century copy

Amazingly, in 1906, another manuscript of Archimedes, TheMethod, was found!



Conclusion

The Method

In this treatise, Archimedes described the preliminary‘mechanical’ investigations he’d conduct to yield hismathematical discoveries

Prior to this 1906 discovery, his claims and proof just ‘worked’– there was no real explanation how he got themIncluded in the text is a note to Eratosthenes, where he notesthat it is easier to supply a proof of a theorem if we first havesome knowledge of what’s involved

So, how did such a manuscript get discovered?



Conclusion

The Method

In 1906, the Danish scholar J. L. Heiberg heard thatConstantinople had a palimpsest2 of mathematical content

Investigations showed 185 pages with Archimedean textcopied in 10th century hand-writing

In the 13th century, attempts had seemingly been made toclear off this text for use as a Euchologion3

Among other works, this contains our only copy of TheMethod

2a manuscript or piece of writing material on which the original writing hasbeen effaced to make room for later writing but of which traces remain.

3one of the chief liturgical books of the Eastern Orthodox and ByzantineCatholic churches



Conclusion

The Method

In 1920, it was still in the Greek Orthodox Patriarchate ofJerusalem’s library in Constantinople

It disappeared in the wake of the Greco-Turkish War in 1922.

Sometime between 1923 and 1930 the palimpsest wasacquired by Marie Louis Sirieix, a ”businessman and travelerto the Orient who lived in Paris.”

It was stored secretly for years in Sirieix’s cellar, where itpalimpsest suffered damage from water and mold.

Sirieix died in 1956, and in 1970 his daughter beganattempting quietly to sell the manuscript.

Unable to sell it privately in 1998 she finally turned toChristie’s – a public auction house in New York – to sell it



Conclusion

The Method

Immediately, the ownership of the palimpsest was contested infederal court

Greek Orthodox Patriarchate of Jerusalem v. Christie’s, Inc.

The plaintiff contended that the palimpsest had been stolenfrom its library in Constantinople in the 1920s.

Judge Kimba Wood decided in favor of Christie’s AuctionHouse

The palimpsest was bought for $2 million by an anonymousbuyer

Donated to the Walters Art Museum in Baltimore, theanonymous buyer funded a high-tech study of the document.



Conclusion

Outline

1 The Era


3 Conclusion



Conclusion

Conclusion

All in all, Archimedes of Syracuse may very well have been thegreatest mathematician of antiquity

Leibniz would write

“He who understands Archimedes and Apollonius willadmire less the achievements of the foremost men oflater times.”

Thus, our next investigation will be one on Apollonius ofPerge.


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