2011 Fall MATH5305
Luis Munoz
Archimedes of Syracuse
Luis Munoz1
The third century BCE saw only a hand full of mathematicians that left an imprint in the history
of mathematics. Archimedes was the greatest in his time. We will discuss his biography, his numerous
discoveries and inventions, along with propositions and his advancement in geometry. Archimedes is
sometimes referred to as a God among men. His mathematical work was superb, and his applications to
weapons and defense stunned the world to this very day.
Archimedes of Syracuse (287-212 BCE) [3] (see Figure 1)
profoundly influenced many regions of study. His influence has been
documented very well. More biographical information about
Archimedes survives than about any other Greek mathematician [3]. Yet,
while a biography of Archimedes was written by his friend Heracleides,
this work was lost, leaving the details about his life obscure. For
instance, it is unknown whether he ever married or had children [7]. It is
probable that he spent time in his youth in Alexandria [3]. Archimedes
died during the Second Punic War, when Roman forces captured the
city of Syracuse after a two-year-long siege [7]. A Roman solider
commanded him to come and meet General Marcus Claudies
Marcellus, but he
declined, saying that
[7]
he had to finish
Figure 1
working
on
the
problem he was devoted to. The soldier was enraged by
this, and killed Archimedes with his sword. The last
words attributed to Archimedes are “Do not disturb my
circles” [7] (See Figure 2).
When Archimedes died, Marcellus was stricken
with the news of the murder. According to history, he
arranged a grand funeral in the necropolis outside the
city’s Achradina gate. A monument was erected on the
burial site, in keeping with Archimedes’ expressed wish,
a stone marker engraved the figure of a sphere
Figure 2[5]
circumscribed by a cylinder; and below, an epitaph
announcing the perfect 2:3 ratio of the surface area of the nested shapes. Archimedes- to some, was an
inventor; to others, an engineer; to the Romans, a mythic opponent; to himself, ever the geometer. The
many roles of Archimedes did not lessen the impression of his work or life. All those who studied his
work are typically called to realize the extent to which his contributions have played a vital role in many
areas of study—even centuries passed.
On April 1, 1776 Sophie Germain was born in France. When she was 13, she heard of the story
of how Archimedes had died. Germain decided that if geometry could hold such fascination for
Archimedes, it was a subject worthy of study [11]. She studied every book about mathematics in her
father’s library even teaching herself Latin and Greek so she could read works like those of Sir Isaac
Newton and Leonhard Euler [11]. Her parents did not like the fact that she had such a fascination with
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Los Fresnos CISD October 25, 2011
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Luis Munoz
mathematics and thought it was inappropriate for a woman to study mathematics. When she was 18, she
obtained lecture notes and began sending her work to Joseph Louis Lagrange [11]. She became the most
famous woman number theorist of her time. Archimedes, even after death, had influenced this little girl
to become a famous mathematician. His dedicated fascination with mathematics has helped to fuel the
persistent study of this field.
Archimedes is universally acknowledged to have been the most proficient mathematician of all
time, on par with the likes of Isaac Newton and Carl Friedrich Gauss. Archimedes derived the
mathematical properties of parabolas, spirals, and polyhedral [1]. His greatest love was mathematics. His
center-of-gravity concept, now a staple of freshman physics, was among the earliest abstractions of
physical objects for the purpose of analyzing nature [1]. Archimedes pioneered the union of mathematics
and physics that was to become a hallmark of modern scientific analysis [1].
The basics of hydrostatics are set down in book 1 of Archimedes’ treatise On Floating Bodies.
Book 2 presents a mathematical tour de force on the tipping tendencies of floating objects, in which
Archimedes deduces to what extent an object can list before it turns topsy-turvy onto the water. Of his
practical works, book 2 rises farthest above the level of mathematical sophistication and physical
ingenuity that prevailed during the classical age. The floatation of ships, sharks, and swimmers is
governed by the tug of war between gravity, which tends to pull an object down, and buoyancy, which
tends to push it up [1]. The force of gravity is exerted between an object and the Earth’s mass-it’s what we
call the object’s weight-and it behaves no differently in water than in air or in a vacuum. The countering
buoyancy force is a consequence of the increase in fluid pressure with depth. Buoyancy arises because the
pressure force pushing up against the bottom of a submerged object is greater than that pushing down
against the top. The pressure forces that act against the object’s sides cancel each other out and don’t
affect the object’s buoyancy. The net result is that buoyancy tries to float an object against gravity’s
tendency to sink it [8].
Archimedes’ contribution to this purely descriptive tale of competing forces was to extent from it
a mathematical kernel that reveals the magnitude of the buoyancy force. His insight is expressed in a
statement since named Archimedes’ Principle: An immersed object is buoyed by a force equal to the
weight of the fluid it displaces [1]. The larger the volume, the more buoyancy the object will experience.
Archimedes’ hydrostatic research is intimately linked with the popular Eureka story involving the
counterfeit crown. According the Vitruvius (a roman architect in the first century BCE) [1], a votive
crown for a temple had been made for King Hiero II, who supplied the pure gold to be used. Archimedes
was asked to determine whether some silver had been substituted by the seemingly dishonest goldsmith.
At the time, melting gold was the manner in which to discover the purity of the metal. Archimedes had to
solve the problem without damaging the crown, so he could not melt it down to a regularly shaped body
in order to calculate its density [7]. Since water is incompressible, the submerged crown would displace an
amount of water equal to its own volume, by dividing the mass of
the crown by the volume of water displaced, the density of the
crown could be obtained [7] (see Figure 3).
Figure 3[9]
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Archimedes must have used a hydrostatic version of a beam
balance, a device that resembles the proverbial scale of justice,
taken from Galileo in his first-ever scientific pamphlet [1]. From one
arm of the beam, Archimedes suspended the suspect crown, and
from the other arm an equivalent weight of pure gold. The two
objects being of equal weight and situated equal distances from the
central pivot, the beam would have been stable in any orientation.
A nudge from Archimedes’ finger would have brought it to its
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Luis Munoz
proper starting orientation-perfectly horizontal [1]. From this point, Archimedes might have immersed the
suspended objects into vessels of water. There were two possible outcomes. If the crown was indeed
made of pure gold, the buoyant forces on it and the matching gold slug would be identical and the balance
would remain horizontal. The other possible outcome is for the adulterated crown to be revealed. Such a
crown would have had a slightly larger volume than one of pure gold because silver or copper takes up
more space than an equivalent weight of gold. Immersed in water, a larger-volume crown would be
buoyed more strongly than the matching slug; the beam of the balance would tip the crown side higher
than the slug side. Evidently the balance did tip, indicating that the artisan had replaced some of the gold
he had been given with a less dense metal [1].
Archimedes dealt with the Law of the Lever principles at the beginning of this treatise Planes in
Equilibrium [3]. Being well trained in Greek geometry, he began by stating seven postulates he would
assume:







Weights which balance at equal distances are equal[10]
Unequal weights at equal distances will not balance but will incline toward the greater weight [10]
Similarly, if anything be taken away from one of the weights, they are not in equilibrium but
incline towards the weight from which nothing was taken [10]
When equal and similar plane figures coincide if applied to one another, their centers of gravity
similarly coincide [10]
In figures which are unequal but similar the centers of gravity will be similarly situated. By points
similarly situated in relation to similar figures I mean points such that, if straight lines be drawn
from them to the equal angles, they make equal angles with the corresponding sides [10]
If magnitudes at certain distances be in equilibrium, (other) magnitudes equal to them will also be
in equilibrium at the same distances[10]
In any figure whose perimeter is concave in (one and) the same direction the center of gravity
must be within the figure [10]
Archimedes has discovered a way to resolve this issue of the crown for King Hiero. There are
many more examples in which Archimedes has developed new principles or methods for handling
different problems. If there was truly a Herculean test of Archimedes’ knowledge of levers and pulleys, it
was the launch of what may have been antiquity’s largest ship, the Syracusia. Commissioned by King
Hieron as a gift for Egypt’s ruler, Ptolemy, the Syracusia weighted in at more than two thousand tons and
was said to have had a library, gymnasium, bath, chapel, promenades lined with flower beds, a
multicolored floor mosaic depicting the entire story of the Iliad, plus accommodations for hundreds of
passengers, soldiers, and horses [1]. Archimedes himself saw to it that vessel was equipped with advanced
weaponry of his own design, as well as an Archimedean screw to bail the bilge [1]. Once built, the
ungainly ship needed a shove into the sea. To the king’s amazement, so the story goes, Archimedes
contrived a mechanical system of compound pulleys, levers,
or screws, with which he single-handedly launched the
vessel with sitting at ease some distance off. Speculation
swirls around the detail of Archimedes’ device and whether
he, in fact, moved the Syracusia itself or another ship or
merely a heavy object which, over time morphed into the
Syracusia [1].
Figure 4[6]
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Like his exploits in ship launching, Archimedes’
invention of the irrigation screw (see Figure 4) that bears his
name is disputed by historians. Some contend that he
merely learned of it while studying in Egypt as a young man.
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Luis Munoz
Yet no such device appears to have existed before Archimedes’ time, and it would have been a reasonable
offspring to his keen interests in geometry and engineering. Motor-driven forms of the Archimedes screw
are employed in modern pumping stations and waste water treatment plants. A tiny version is also found
in mechanical cardiac assist system, which maintains blood flow in patients with heart failure or
undergoing heart surgery. The screw conveyor, a variation of Archimedes screw, is commonly used in
industry to move grains, powders, and other bulk materials through troughs or pipes [1]. The screw pump
was simple and elegant; it had a revolving screw-shaped blade inside a cylinder and could be turned by
hand [6]. In the third world countries it is still a preferred way to irrigate agricultural fields without
electrical pumps [6].
Another fascinating invention of this brilliant
mathematician was the Claw of Archimedes, a “ship
shaker” (see Figure 5). It was used to defend the city of
Syracuse. The claw worked like a modern day crane with
metal grappling hooks [6]. When the claw dropped onto an
attacking ship the arm would swing upwards, lifting the
ship out of the water and possibly sinking it. There have
been modern experiments to test the feasibility of the claw,
and in 2005 a television documentary entitled Super
weapons of the Ancient World built a version of the claw
and concluded that it was a workable device [7].
Also in the line of military defense, Archimedes is
said to have set ships afire by having his soldiers focus
sunlight onto the vessels from banks of mirrors (see Figure
6). To the best of our current day knowledge, this is almost
Figure 5[6]
certainly untrue. The popular Discovery Channel program
MythBusters tested a modern version of the supposed weapon, to no effect; however MIT was able to
ignite a stationary wooden target with its own Archimedean
“Death Ray.” The experiment was with 127 one-foot square
mirror tiles, focused on a mock wooden-ship at a range of around
100 feet [7]. More to reasonably, the Syracusans might have used
polished metal plates to reflect the morning sun into the eyes of
Roman sailors approaching from the East [1]. With all of these
varying factors to consider, the “Death Ray” one cannot truly say
whether or not it may have actually worked.
Archimedes enjoyed the greatest fame in antiquity,
however, for his design of various engines of war [3]. These
engines enabled Syracuse to hold off the Roman siege for many
months. Archimedes devised various missile launchers as well as
huge cranes by which he was able to lift Roman ships out of the
water and dash them against the rocks or simply dump out the
crew [3]. In fact, he was so successful that any time the Romans
saw a little rope or piece of wood come out from the walls of the
city, they fled in panic [1].
Figure 6 [7]
Among his numerous war machines, Archimedes, in a
brief treatise titled Measurement of the Circle, listed numerical results unlike anything found before his
time. Its first proposition, in addition, gives Archimedes’ answers to the question of squaring the circle,
by showing that the area of a circle of given radius can be found once the circumference is known. The
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Luis Munoz
third proposition of this treatise complements the first by giving a numerical approximation to the length
of the circumference [3].
Proposition 1 The Area of any circle is equal to the area of a right triangle in which one of the
legs is equal to the radius and the other to the circumference [3].
Archimedes gave a rigorous proof, using Eudoxian exhaustion. Namely, if K is the area of the given
triangle, Archimedes first supposed that A>K. By inscribing in the circle regular polygons of
successively more sides, he eventually determined a polygon of area P such that A-P<A-K. Thus, P>K.
Now the perpendicular from the center of the circle to the midpoint of a side of the polygon is less than
the radius, while the perimeter of the polygon is less than the circumference. It follows that P<K, a
contradiction. Similarly, the assumption that A<K leads to another contradiction and the result is proved
[3]
.
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Proposition 3 The ratio of the circumference of any circle to its diameter is less than 3 but greater than
10
3 [3].
71
Archimedes’ proof of this statement provided algorithms for determining the perimeter of certain regular
polygons circumscribed about and inscribed in a circle. Namely, Archimedes began with regular
hexagons, the ratios of whose perimeters to the diameter of the circle are known from elementary
geometry. He then in effect used the following lemmas (given in modern notation) to calculate, in turn,
the ratios to the diameter of the perimeters of regular polygons with 12, 24, 48, and 96 sides, respectively.
Lemma 1 Suppose OA is the radius of a circle and CA is tangent to the circle at A. Let DO bisect angle
COA and interest the tangent at D. Then DA/OA = CA/ (CO+OA) and DO2=OA2+DA2 [3] (See Figure 7).
Lemma 2 Let AB be the diameter of a circle and ACB a right triangle inscribed in the semicircle. Let
AD bisect ∠𝐶𝐴𝐵 and meet the circle at D. Connect DB. Then AB2/BD2=1+(AB+AC)2 and AD2=AB2BD2 (see Figure 7).
Archimedes used the first lemma repeatedly to develop a recursive algorithm for determining the desired
ratio using circumscribed polygons. He began
by assuming that ∠𝐶𝑂𝐴 is one-third of a right
angle (30°), so CA is half of one side of a
circumscribed regular hexagon. Therefore,
CA and CO are known. Since ∠𝐷𝑂𝐴 =15°, it
follows that DA is half of one side of a regular
12-gon. DA and DO are then calculated by
use of the lemma. Next, ∠𝐷𝑂𝐴 is bisected to
1
get an angle 7 2 °. The piece of the tangent
subtending that angle is then half of one side
Figure 7 [3]
of the regular 24-gon. Its lengths can be
calculated as well. If r is the radius of the circle, 𝑡𝑖 half of one side of a regular 3𝑥2𝑖 -gon (𝑖 ≥ 1), and 𝑢𝑖
the length of the line from the center of the center of the circle to a vertex of that polygon, the lemma can
be translated into the recursive formulas.
𝑟𝑡
𝑖
𝑡𝑖+1 = 𝑢 +𝑟
,
𝑖
2
𝑢𝑖+1 = √𝑟 2 + 𝑡𝑖+1
The ratio of the perimeter of the ith circumscribed polygon to the diameter of the circle is then
6(2𝑖 𝑡𝑖 : 2𝑟: 3(2𝑖 𝑡𝑖 ): 𝑟.
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Luis Munoz
Archimedes developed a similar algorithm for inscribed polygons by use of the second lemma, and in
both cases, provided explicit numerical results at each state. For example, in his calculations involving
hexagons in both the circumscribed and inscribed cases, he needed to evaluate the ratio √3: 1. What he
wrote indicates that he knew that this ratio is greater than 265:163 and less than 1351:780. Although it is
not known exactly how Archimedes found these results, it is certain that he, like many great
mathematicians of later times, was a superb calculator. After four steps of both algorithms, in fact, he
concluded that the ratio of the perimeter of the circumscribed 96-sided polygon to the diameter is less
1
than 14,688: 4673 2 = 3 +
1
2
1
4673
2
1
667
1
< 3 7, while the ratio of the inscribed 96-sided polygon to the diameter
10
is greater than 6336: 2077 4 > 3 71, thus proving the theorem [3].
Archimedes’ proof is the first recorded method for actually computing π [3]. Once the method was known,
it was merely a matter of patience and persistence to calculate π to as great a degree of accuracy as
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desired. Archimedes does not tell us why he stopped at 96-sided polygons. But his value of 3 has
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become a standard approximation for π to the present day [3].
Archimedes had a favorite proof. It is the same one he requested to be engraved on his headstone.
The proof was consisting of a sphere and a cylinder of the same height and diameter. Archimedes had
proven that the volume and surface area of the sphere are two thirds that of the cylinder including its
bases [2]. This is just one of the many contributions Archimedes was able to accomplish in his lifetime. In
the end, his extreme devotion to mathematics is what killed him. If he had moved out of the soldier’s way,
or tried to explain his reasoning behind his request, maybe he would have lived to contribute more.
Archimedes was a man with great thought. He lived forever in his own world, and was not able
to think about the day to day worries like bathing or eating. This extreme devotion has left a deep
impression which moves those inspired by his work and story to continue in his work. For this reason, his
work will always be regarded as the best in his time, and never lose relevance. It is astonishing what
Archimedes was able to think of and to accomplish in his short life. He was one of the very first
mathematicians to really break the boundaries with his revolutionary ideas and work.
Reference
[1] Hirshfeld, A., Eureka Man, Walter and Company, 2010
[2] Heath, T., The Works of Archimedes, Cambridge, 1897
[3] Katz, V. J., A History of Mathematics, Pearson, 2009
[4] http://www.mlahanas.de/Greeks/Archimedes.htm (retrieved Oct. 22, 2011)
[5] http://www.cartoonstock.com/directory/a/archimedes.asp (retrieved Oct. 22, 2011)
[6] http://www.the-business-of-patents.com/archimedes-inventions.html (retrieved Oct. 22, 2011)
[7] http://en.wikipedia.org/wiki/Archimedes (retrieved Oct. 22, 2011)
[8] http://en.wikipedia.org/wiki/Buoyancy (retrieved Oct. 22, 2011)
[9] http://math.nyu.edu/~crorres/Archimedes/Crown/CrownIntro.html (retrieved Oct. 22, 2011)
[10] http://ebooks.cambridge.org/chapter.jsf?bid=CBO9780511695124&cid=CBO9780511695124A034
(retrieved Oct. 22, 2011)
[11] http://en.wikipedia.org/wiki/Sophie_germain (retrieved Oct. 22, 2011)
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