Archimedes of Syracuse

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Rebecca Imogene Rogers
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2 Table of contents 1 2 3

3 Outline 1 2 3

4 The Punic Wars

5 The Punic Wars The Punic 1 Wars were a series of wars between Carthage and Rome, occurring between 264 BCE and 146 BCE The First Punic War: 264 BCE BCE The Second Punic War: 218 BCE BCE The Third Punic War: 149 BCE BCE By 146 BCE, the Romans had completely defeated the Carthaginians 1 The term Punic comes from the Latin word Punicus, meaning Carthaginian referencing the Carthaginians Phoenician ancestry.

6 Siege of Syracuse In 214 BCE, in the middle of the Second Punic War, Rome attempted to capture the Kingdom of Syracuse The Roman forces, led by General Marcellus, laid siege to the island for three years Syracuse, known for its fortifications and great walls, was almost impenetrable Defending his city, Archimedes did all he could to fend off the Roman invaders In particular, he used his knowledge of pulleys and levers to invent war machines like catapults and devices that could set fire to Roman ships from afar

7 Siege of Syracuse Thomas Ralph Spence. Archimedes directing the defenses of Syracuse

8 Siege of Syracuse In 212 BCE, the Romans managed to invade the city Despite direct orders from Marcellus to spare Archimedes, he was slain by a Roman soldier Archimedes was reportedly 75 at the time of his death, and thus was born in 287 BCE

9 Siege of Syracuse Thomas Degeorge. Death of Archimedes

10 Domenico Fetti. Archimedes Thoughtful

11 Respect for Archimedes talent and accomplishments persisted throughout the centuries after his death. In the 18th century, the philosopher Voltaire would write: There was more imagination in the head of Archimedes than in that of Homer. We now take a look at some of his work and discuss his contributions to mathematics...

12 Outline 1 2 3

13 On the Equilibriums of Planes The Law of the Lever: Archimedes further developed what is now called the Archimedean axiom of symmetry: Bilaterally symmetric bodies are in equilibrium Generalizing this static principle, Archimedes developed a general understanding of levers About a century earlier, Aristotle had published Physics, but his discussion of the lever was developed via speculation and non-mathematical reasoning Archimedes developed it much like Euclid s Elements, via postulates and deductive reasoning.

14 On the Equilibriums of Planes Another physics treatise, On Floating Bodies, was a two-book work on fluids and buoyancy. In it, he noted the following: Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. In De Architectura, a book written by the Roman architect Marcus Vitruvius Pollo some time during the first century BCE, a fun story involving Archimedes was told about how his intellect ousted a fraudulent blacksmith Much of Archimedes work in these two works laid the foundation for the mathematical underpinning of physics For his contributions to physics and its mathematical justification, he is sometimes referred to as the Father of Mathematical Physics.

15 Sand-Reckoner Recall that the ancient Greek mathematicians drew a sharp distinction between logistic and arithmetic In particular, logistic was somehow looked down upon In the mid-third century BCE, Aristarchus of Samos proposed a heliocentric model of the universe Archimedes would later note, His [Aristarchus ] hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun on the circumference of a circle, the Sun lying in the middle of the orbit, and that the sphere of fixed stars, situated about the same center as the Sun, is so great that the circle in which he supposes the Earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface.

16 Sand-Reckoner In his work Psammites (or, Sand-Reckoner), Archimedes claimed he could write down a number greater than the number of grains of sand required to fill Aristarchus universe His computation led to an estimate of grains Recall that it was around the third century BCE that Greek numeration switched to the Ionian system with 27 distinct letters With no positional notation, this system only counted up to a myriad M = myriad = 10, 000 = 10 4 Archimedes work led to him extending this numeration system in an incredible way A clear contribution to logistic

17 Sand-Reckoner Archimedes extended naturally to a myriad-myriad, essentially a word for the value Working in this base, he defined orders in the following way: For a given number x, it was in order n if: 10 8(n 1) x < 10 8n where x is order 1 if x < He did this up until a myriad-myriad of orders: up to Then he defined values less than to be of period one. Continuing, he defined values up to a myriad-myriad of periods The largest value he wrote down was essentially ( (10 8 ) (108 ) ) (10 8 ) =

18 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 a hexagon) and let P n denote its perimeter. Circumscribe the circle with a regular n-gon and let p n denote its perimeter. Now, do the same for a 2n-gon:

19 Measurement of the Circle It turns out that P 2n and p 2n can be found recursively: P 2n = 2p np n p n + P n and p 2n = p n P 2n. His estimates for square roots utilized the Babylonian algorithm

20 Measurement of the Circle Similar recursive formulae exist for the areas of each polygon. Denoted A n and a n respectively for the area of the inner n-gon and outer n-gon Starting with the hexagon, Archimedes carefully computed each iteration until he had approximated P 96, p 96, A 96 and a 96 From here he provided bounds for π: < π < This estimate was better than both the Egyptians and the Babylonians

21 On Spirals Archimedes was also enchanted by the problems of Greek antiquity In an effort to trisect the angle, he investigated the Archimidean-spiral In modern, polar terms: r = aθ Of note: Archimides himself attributed the spiral to a friend of his, Conon of Alexandria

22 On Spirals To trisect an angle: Interestingly, the spiral can also be used to square the circle.

23 Quadrature of the Parabola Conic sections had been known for almost a century by Archimedes time, yet no one had attempted to find their areas In Proposition 17 of Quadrature of the Parabola, he provided a construction to compute the area of any segment of a parabola: Start by taking an arbitrary parabola:

24 Quadrature of the Parabola In the preamble of this treatise, we find the Axiom of Archimedes: That the excess by which the greater of two unequal areas exceeds the less can, by being added to itself, be made to exceed any given finite area. This axiom, sometimes referred to as the archimedean property of R, is an absolute staple For example, it is found in part (a) of Theorem 1.20 in Rudin s Principles of Mathematical Analysis

25 On Conoids and Spheroids Among other things, this treatise contains a proof that the area of the ellipse given by x 2 + y 2 = 1 a 2 b 2 is equal to πab. Unlike the parabola, where he computed the area of any segment of the curve, Archimedes was unable to do the same for the ellipse and hyperbola Through the modern lens, we see that the integral necessary to compute the area of parabolic segments involves only polynomials The respective integrals for ellipses and hyperbolas, however, involve transcendental functions.

26 On the Sphere and Cylinder Archimedes himself seemed most enchanted by this particular treatise In fact, he asked the following to be carved on his tombstone:

27 On the Sphere and Cylinder This is due to the following discovery: Given, It follows that, V cylinder V sphere = SA cylinder SA sphere = 3 2. Archimedes claimed that this fact was unknown before him.

28 On the Sphere and Cylinder In Book II of this treatise, Archimedes gave several computations on how to cut a sphere such that: The two parts surface areas were in a desired ratio with one another The two parts volumes were in a desired ratio with one another This question was much more difficult. It is theorized that these constructions were inspired by Euclid s Division of Figures

29 Book of Lemmas In this treatise, he tackled basic problems that were not categorized as higher math For example, the quadrature of arbelos of the shoemaker s knife:

30 Book of Lemmas Archimedes showed that the area of the arbelos is equal to that of the circle PRCS Another example given in his Book of Lemmas is another construction for trisecting an angle.

31 The Method Recall that Euclid s Elements survived in many Greek and Arabic manuscripts, Archimedes treatises persist via a single Greek original from the 1700s, copied from a 9th or 10th century copy Amazingly, in 1906, another manuscript of Archimedes, The Method, was found!

32 The Method In this treatise, Archimedes described the preliminary mechanical investigations he d conduct to yield his mathematical discoveries Prior to this 1906 discovery, his claims and proof just worked there was no real explanation how he got them Included in the text is a note to Eratosthenes, where he notes that it is easier to supply a proof of a theorem if we first have some knowledge of what s involved So, how did such a manuscript get discovered?

33 The Method In 1906, the Danish scholar J. L. Heiberg heard that Constantinople had a palimpsest 2 of mathematical content Investigations showed 185 pages with Archimedean text copied in 10th century hand-writing In the 13th century, attempts had seemingly been made to clear off this text for use as a Euchologion 3 Among other works, this contains our only copy of The Method 2 a manuscript or piece of writing material on which the original writing has been effaced to make room for later writing but of which traces remain. 3 one of the chief liturgical books of the Eastern Orthodox and Byzantine Catholic churches

34 The Method In 1920, it was still in the Greek Orthodox Patriarchate of Jerusalem s library in Constantinople It disappeared in the wake of the Greco-Turkish War in Sometime between 1923 and 1930 the palimpsest was acquired by Marie Louis Sirieix, a businessman and traveler to the Orient who lived in Paris. It was stored secretly for years in Sirieix s cellar, where it palimpsest suffered damage from water and mold. Sirieix died in 1956, and in 1970 his daughter began attempting quietly to sell the manuscript. Unable to sell it privately in 1998 she finally turned to Christie s a public auction house in New York to sell it

35 The Method Immediately, the ownership of the palimpsest was contested in federal court Greek Orthodox Patriarchate of Jerusalem v. Christie s, Inc. The plaintiff contended that the palimpsest had been stolen from its library in Constantinople in the 1920s. Judge Kimba Wood decided in favor of Christie s Auction House The palimpsest was bought for $2 million by an anonymous buyer Donated to the Walters Art Museum in Baltimore, the anonymous buyer funded a high-tech study of the document.

36 Outline 1 2 3

37 All in all, may very well have been the greatest mathematician of antiquity Leibniz would write He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times. Thus, our next investigation will be one on Apollonius of Perge.

Archimedes Quadrature of the Parabola

Archimedes Quadrature of the Parabola Archimedes and the Quadrature of the Parabola MATH 110 Topics in Mathematics Mathematics Through Time November 1, 2013 Outline Introduction 1 Introduction 2 3 Who was Archimedes? Lived ca. 287-212 BCE,