The Era Works of Archimedes Conclusion
Archimedes of Syracuse
Douglas Pfeffer
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion Table of contents
1 The Era
2 Works of Archimedes
3 Conclusion
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion Outline
1 The Era
2 Works of Archimedes
3 Conclusion
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion The Punic Wars
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion The Punic Wars
The Punic1 Wars were a series of wars between Carthage and Rome, occurring between 264 BCE and 146 BCE The First Punic War: 264 BCE - 241 BCE The Second Punic War: 218 BCE - 201 BCE The Third Punic War: 149 BCE - 146 BCE By 146 BCE, the Romans had completely defeated the Carthaginians
1The term Punic comes from the Latin word Punicus, meaning ‘Carthaginian’ – referencing the Carthaginians’ Phoenician ancestry. Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion Siege of Syracuse
Thomas Ralph Spence. Archimedes directing the defenses of Syracuse. 1895.
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion Siege of Syracuse
Thomas Degeorge. Death of Archimedes. 1815.
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion Archimedes of Syracuse
Domenico Fetti. Archimedes Thoughtful. 1620.
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion Archimedes of Syracuse
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...
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion Outline
1 The Era
2 Works of Archimedes
3 Conclusion
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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.
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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.
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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.”
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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 1063 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 0 = 10, 000 = 104 Archimedes’ work led to him extending this numeration system in an incredible way A clear contribution to logistic
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion Sand-Reckoner
Archimedes extended naturally to a myriad-myriad, essentially a 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 108·108 to be of period one. Continuing, he defined values up to a myriad-myriad of periods The largest value he wrote down was essentially
8 8 (10 ) 16 (108)(10 ) = 108·10 .
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes 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 a hexagon) and let Pn denote its perimeter. Circumscribe the circle with a regular n-gon and let pn denote its perimeter. Now, do the same for a 2n-gon:
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion Measurement of the Circle
It turns out that P2n and p2n can be found recursively:
2pnPn p P2n = and p2n = pnP2n. pn + Pn His estimates for square roots utilized the Babylonian algorithm
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion Measurement of the Circle
Similar recursive formulae exist for the areas of each polygon.
Denoted An and an 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 P96, p96, A96 and a96 From here he provided bounds for π:
10 10 3 71 < π < 3 70 This estimate was better than both the Egyptians and the Babylonians
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion On Spirals
To trisect an angle:
Interestingly, the spiral can also be used to square the circle.
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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:
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion On Conoids and Spheroids
Among other things, this treatise contains a proof that the area of the ellipse given by
x2 y 2 a2 + b2 = 1 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.
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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:
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion On the Sphere and Cylinder
This is due to the following discovery: Given,
It follows that, V SA 3 cylinder = cylinder = . Vsphere SAsphere 2 Archimedes claimed that this fact was unknown before him.
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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:
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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.
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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!
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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?
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion The Method
In 1906, the Danish scholar J. L. Heiberg heard that Constantinople had a palimpsest2 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 Euchologion3 Among other works, this contains our only copy of The Method
2a 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. 3one of the chief liturgical books of the Eastern Orthodox and Byzantine Catholic churches Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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 1922. 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
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion 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.
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion Outline
1 The Era
2 Works of Archimedes
3 Conclusion
Douglas Pfeffer Archimedes of Syracuse The Era Works of Archimedes Conclusion Conclusion
All in all, Archimedes of Syracuse 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.
Douglas Pfeffer Archimedes of Syracuse