Kimberly Hannusch
History of Math
Essay 2
Archimedes: The Pioneer of Modern Engineering
In our modern day, where planes are commonplace, nearly everyone living in a first
world country owns a car, and most things are mass-produced by manufacturing robots, very
few marvel at the mathematics and engineering required to make these complex machines
work. Most people can’t even picture a time when there were no machines, much less begin to
consider how they could be invented from scratch. However, the majority of people would
agree that the person who began to envision the earliest contraptions – when there were no
precursors off of which to model – must have been a brilliant thinker. They would not be
wrong. Many of the simplest mechanical tools which have since evolved into our machines
today can be traced back to the ingenious mind of the famous mathematician named
Archimedes. He can arguably be considered the pioneer of modern mechanical engineering.
Archimedes was born in Syracuse in 287 B.C. He was the son of an astronomer named
Phidias, who may have passed down his interest in mathematics to his son. Little else is known
of his early life, though it is known that he went to Alexandria as a youth to study mathematics
at the school which Euclid established. Because of his fascination with mathematics and his love
for solving difficult problems, he thrived in the university setting. It is said of him that he would
forget to eat, drink, and sleep when he was trying to think through tough puzzles. Also, if a
problem came to him at an inconvenient time or place, he would stop whatever he was doing
and use anything available to draw it out, including sticks scratched in the dirt.
After exhausting the resources available to him in Alexandria, he returned to Syracuse
and continued his self-guided learning there. He formed a friendship with King Hiero II of
Syracuse and solved many problems for him. For example, the king had a gold crown made for
himself, but he was suspicious that the maker had only plated it in gold rather than making it
entirely out of gold. He asked Archimedes to help him find out if it was pure gold without
damaging or altering the crown in any way. It is said that Archimedes discovered the answer to
this problem when he was taking a bath because he noticed the change in water level as he got
into the tub. He realized that he could measure the displacement of water by the crown and an
equal mass of pure gold and compare the two. He was so excited when he discovered the
answer to this problem, that he immediately jumped out of the bath shouting, “Eureka!” and
ran to tell the king while stark naked.
Archimedes’ famous discoveries and developments can be divided into three main
categories: proofs of geometric theories, mechanical solutions for static and hydrostatic
problems, and counting methods such as the method of exhaustion. His geometric proofs
included his discovery that a sphere circumscribed in a cylinder
is two-thirds of the cylinder’s volume (see figure at right). He
was so proud of this discovery that he asked that his gravestone
be engraved with the drawing of it when he died. He found
many other discoveries about areas and volumes of spheres,
cylinders, parabolas, and even spirals. In terms of mechanical static and hydrostatic problems,
he was noted for his ideas on buoyancy. In fact, in his work On Floating Bodies, he develops the
theorem for finding the weight of objects floating in liquids, which is now called the Archimedes
principle. He also developed theorems for finding the center of gravity of figures in a plane. His
counting methods include his approximation of pi, where he used the method of exhaustion to
bound pi between 22/7 and 223/71, a fairly accurate approximation. He also approximated
square roots and estimated the number of grains of sand which could be fit into the entire
universe (which was a difficult feat considering the bulky Roman numerals and the sheer size of
the numbers with which he was working).
Perhaps some of the most colorful stories about Archimedes come from his role in the
defense of his city against the invading Romans during the Second Punic War. During this
period, his mechanical and engineering genius shone brightly. After the death of King Hieron II,
Syracuse was attacked by the Romans. The successor to the king, Hieronymus, recruited
Archimedes for the defense of the town. Of course, it wasn’t his brawn or military strength that
was needed; rather, his engineering mind was put to work. He invented catapults to launch
large rocks at advancing ships, pulleys to lift boulders which were then dropped on ships as
they sailed underneath, and parabolic mirrors which intensified
the sun’s rays onto ships and caught them on fire (see picture at
right). It is even said that he developed a crane system which
would pick up ships off of the water and then dash them against
the cliffs or let them drop again into the water, breaking upon
impact. Because of these masterful inventions, the Romans were
kept at bay for quite a while. Finally, the Roman general Marcellus laid off the offensive and
started a siege which eventually caused the defeat of Syracuse.
When Syracuse was captured, Archimedes was killed in the chaos. It is said that
Marcellus ordered that Archimedes be spared since such a brilliant mind should not go to
waste. However, the tale goes that Archimedes did not notice when the city was taken since he
was absorbed in a mathematical problem at the time. When a Roman soldier told him to come,
he completely ignored him. This angered the soldier, who ran him through with his sword, not
realizing that the man he just killed was Archimedes. On that day, the ancient world lost a
brilliant thinker and engineer. Not only was he able to solve intense theoretical math problems,
but he also took it one step further than other classical Greeks were willing to do: he
experimented and applied theoretical math to real world problems.
It is obvious from the accounts of Archimedes that he was a brilliant man who was
enthralled with mathematics and the ways that it could be applied to problems in his daily life.
His jump from the Greeks’ purely theoretical math to the more experimental, physical math
would pave the way for scientists in the future such as da Vinci, Newton, and Leibniz. From his
insistence on pure mathematics’ practical usefulness, later mathematicians would be able to
develop more concrete laws and theories which drive our modern world to this day. So the next
time you start up your car, fly on a plane, or see structures being built by large machinery, you
can be thankful for the wonderful mind of Archimedes which turned ancient mathematics on its
head towards a more practical direction.
Sources:
Hoyt, Lloyd. "Archimedes." Archimedes. N.p., n.d. Web. 17 June 2016.
<http://www.math.wichita.edu/history/men/archimedes.html>.
O'Connor, J. J., and E. F. Robertson. "Archimedes of Syracuse."Biographies. University of St.
Andrews, Jan. 1999. Web. 17 June 2016. <http://www-groups.dcs.stand.ac.uk/~history/Biographies/Archimedes.html>.
"The History of Archimedes." The Archimedes Palimpsest. Palimpsest Conservation, n.d. Web. 17
June 2016. <http://archimedespalimpsest.org/about/history/archimedes.php>.
Violatti, Cristian. "Archimedes." Ancient History Encyclopedia. Ancient History Encyclopedia, 24
June 2013. Web. 17 June 2016. <http://www.ancient.eu/Archimedes/>.
All pictures from:
https://en.wikipedia.org/wiki/Archimedes