Heather Hatlo

Professor Walton

Math G

December 1, 2003

 

The Mathematics of Archimedes

Archimedes is considered by most historians to be one of the greatest mathematicians of all time. He is also thought of as one of the greatest scientists of all time, known for many important inventions. His fields of study included; hydrostatics, buoyancy, static machines, and pycnometry (the measurement of the volume or density of an object). He has been nicknamed the ìfather of integral calculus,î for the tremendous impact he had on the study of integration and calculus. Archimedes was instrumental in assisting the armies of his homeland to defend themselves against the Romans. His war machines helped avoid invasion for many years and in turn earned him praise throughout the region. While he was an extremely famous inventor in his day, he took much more pride and pleasure in pure mathematics. He would get so engrossed in solving mathematical equations that it was common for him to skip meals and neglect to bathe for days at a time. It wasnít until after his death that he really began to receive recognition for his mathematic contributions. Even without recognition during his life, Archimedes preferred to spend his days theorizing rather than inventing anything tangible.

Archimedes was born in 287 B.C. in Syracuse, Sicily. His father, Phidias, was an astronomer. This had an important impact on Archimedesí future studies. He was fascinated by his fatherís work and began learning about Astronomy at a very young age. Having educated parents gave Archimedes an edge over his peers. His parents insisted that he study and attend school; so he traveled to Alexandria, Egypt as a young man and studied under the followers of Euclid. He had a very inquiring mind and delved into many fields of study when he went away to school. Another advantage Archimedes was born into was his relationship with King Heiron II. Although there is not specific evidence that this relationship contributed to his rise as an important figure of his time, it is very likely that in a monarchal society it did have some affect on Archimedesí stature in the community. While we do not know if this relationship was familial or just friendly, we do know it was an important relationship to Archimedes because of book dedications he made to members of the royal family, and prefaces he wrote in his book.

The greatest contribution Archimedes made to the mathematical world was his work in the study of integration. He was fascinated with discovering volumes and the areas of many objects. He found ways to calculate the areas of difficult to measure shapes, like parabolas and ellipses. To discover these measurements, Archimedes broke up images into an infinite number of rectangles and then added the areas of the sections together. An example of this can be seen in the image on the right. This is known as the method of exhaustion; ìa technique of approximating the area of a region with an ever increasing number of polygons, with the approximations improving at each step and the exact area being attained after an infinite number of these stepsî (Guide to History of Calculus). Specifically, for solving the quadrature of a parabola, he first approximated the area with a large number of triangles and then used a ìdouble reductio ad absurdum argumentî (Guide to History of Calculus) to prove his result. Archimedesí work in integration would provide the foundation for the development of calculus later in history.

Integration wasnít the only area of mathematics that Archimedes made huge advances in. Archimedes was particularly fascinated with geometric applications in astronomy, which is not surprising considering his fatherís profession. He was the first person to calculate the length of a year, after building a machine to measure the angles of the rising sun. He made many other significant contributions to the field of geometry; among the most important was his discovery of a very accurate approximation of PI. He determined it to be somewhere between 3.1408 and 3.1428. He computed this as follows:

With respect to a circle of radius r, let

Further, let denote the regular inscribed polygons, similarly, for the circumscribed polygons. The following formula gives the relation between the perimeters and areas of these polygons.

 

 

Using n-gons up to 96 sides he was able to derive:

 

         Calculations from Texas A&M Math Dept.

 

To describe this in simpler terms, Archimedes found the approximate measure of PI by drawing polygons within a circle. Then he determined ìthe length of the perimeter of a polygon inscribed within a circle (which is less than the circumference of the circle) and the perimeter of a polygon circumscribed outside a circle (which is greater than the circumference)î (PBS, NOVA series).  Although there had been many close approximations of PI prior to Archimedesí discovery, this was the first time it had been calculated theoretically rather than actually physically measured. It was also the most accurate measurement at the time.  Even to this day, with super computers measuring PI out to one trillion decimal places, Archimedesí calculation can be used for most practical applications. However important the calculation of PI has been to the world, Archimedes considered a different calculation to be his most important in geometry. He proved that the volume of a sphere is two-thirds the volume of a cylinder that circumsribes the sphere. He felt this was so important in fact, that he requested an illustration of this be inscribed on his tombstone.

            With all of the time and effort Archimedes spent on solving math equations and analyzing complex problems, he soon found he could solve a number of practical problems by putting his mathematical skills to good use. He was often called on by the King to solve problems for the monarchy. One of the issues he resolved for King Heiron II, is known as the Gold Crown problem. The King had hired a goldsmith to make him a crown of pure gold. However, upon delivery of the crown, the King suspected some silver had been used. He called on Archimedes to help prove this. After spending some time thinking this issue over, Archimedes discovered a simple way to determine the crownís composition while he soaked in a bathtub one afternoon. He realized that the displaced water in the tub was proportional to the weight of the object submerged in the water. This observation is now known as Archimedesí Principle, and it opened up the door to the study of hydrostatics and buoyancy. To prove the crown was indeed made with some silver, Archimedes took an amount of gold that should have been equal to the amount in the crown. Placing them both in a tub of water, it was clear that the crown did not displace as much water, and this proved the fraudulent work of the goldsmith. The King was so pleased with Archimedes, that he requested his assistance in the fight against the Romans. Archimedes invented a number of machines and devices that could be used to defend Sicily during the First and Second Punic Wars.

            The weapons that Archimedes invented for war fall into three main categories: a) cranes or claws, b) catapults of every size and description, and c) mirrors that focused sunlight on ships and set them alight.  Probably the most talked about of these inventions, was his system of mirrors that could be focused on enemy ships in the harbor. These mirrors would then reflect rays of sunlight directly on the ships and cause them to catch fire and burn. This invention became legendary, and for a long time scholars  debated whether or not such an invention could have been possible in the time of Archimedes.  For centuries it was believed that Archimedesí mirrors were only a myth. That was until an engineer proved it was possible in 1973. Using 70 copper plated glass lenses, Ioannis Sakkas conducted an experiment on the island of Salamina. He placed the lenses in a circle and focused the suns rays on a small boat drifting 55 feet out in the water. The boat was crafted in a similar fashion to the Roman ships of Archimedesí day. Within moments the boat was engulfed in flames, proving Archimedesí invention had worked. Other important inventions used in the punic wars include; cranes that lifted enemy ships out of the water and dashed them against the rocks, and catapults that hurled bolts and stones varying distances.

            The successful use of Archimedesí catapults was due to his great understanding of how levers and pulleys operate. He originally created a system of levers to assist the King with more traditional work in the kingdom, but they proved very useful as weapons of defense. His first description of the lever was around 260BC.  He applied this understanding of levers to create remarkable catapults that could toss quarter-ton stones at Roman ships in the harbor. Archimedes is quoted as saying; ìGive me a lever long enough, and a place to stand, and I will move the worldî (Archimedes, 230BC). This statement has become an inspiration to millions of inventors, engineers and scientists alike.

            Along with other uses for his system of levers and pulleys, many of Archimedesí inventions were not used only for war. He enjoyed solving practical problems with his inventions, like the keeping of time for example. During his life,  the Greeks had a very good system of tracking the hours in the day, but once the sun went down, and they could not follow its position in the sky, they had trouble marking the time. The changing of the seasons also made it difficult for people to track time, since days were longer and shorter at different points of the year. Archimedes solved this problem with his creation of a more advanced water clock. His new device allowed for accurate calculations of time during the day and night, with a margin of error of only two minutes. While water clocks had existed for some time, Archimedes added gears and also showed the movement of the planets and the orbiting moon. Another signifigant invention was Archimedesí Screw. He invented this device around 269BC, and it was used to transport water with only a small amount of labor. It was similar in shape to a screw used to hold wood together, and it could be used as a pump to remove water from the hold of large ships. One form of this screw consisted of a circular pipe enclosing a helix, and then inclined at a forty-five degree angle to the horizontal with the lower end dipped into the water. When you rotate the device, the water rises into the pipe. This invention was so important, that it is still used in the present day in countries around the world for irrigation of crops and in water treatment centers. The picture on the left is ìone of eight 12-ft.-diameter Archimedesí screws used to handle rainstorm runoff in Texas City, Texas. Each screw is driven by a 750-hp diesel engine and can pump up to 125,000 gallons per minuteî (Drexel University).

            We are fortunate to know so much about Archimedesí inventions and mathematical calculations because of the many literary works he left behind. These books have proved to be invaluable in the study of mathematics. Some of these important works include The Quadrature of the Parabola; in which Archimedes calculated the area of a segment of a parabola, as was discussed earlier in this paper. Archimedes wrote two volumes of On Floating Bodies; books that detailed the law of equilibrium of fluids and proved that water around a center of gravity will take a spherical form.  Another important contribution came from the book The Sand Reckoner. It was in this that ìArchimedes proposed a number system capable of expressing numbers up to 8x10¹⁶ in modern notations. He argued in this work that this number is large enough to count the number of grains of sand which could be fitted into the universe. There are also important historical remarks in this work, for Archimedes has to give the dimensions of the universe to be able to count the number of grains of sand which it could containî (University of St. Andrews).  The book On the Sphere and Cylinder contained the works that Archimedes was most proud of; the discovery that the area and volume of a sphere are in the same relationship as the area and volume of a circumscribed cylinder.

            His most recent work, The Method, was only discovered in the early nineteen hundreds. It was discovered in a larger set of documents, known as The Archimedes Palimpset. An image of these documents is shown on the page above. The Method has been an important insight into the mind of Archimedes. It has helped give mathematicians and scientists an understanding of the process Archimedes went through as he made some of his most important discoveries. He detailed the way he measured a variety of geometric figures against one another, and calculated the volumes and areas with those measurements. Although Archimedes did not rely on the information in The Method as proof of his calculations; (instead he preferred to publish work done with the method of exhaustion), this book has proved to be great research material for those studying Archimedesí work. Other well known literary works of Archimedes include; On Plane Equilibriums, On Spirals, On Conoids and Spheroids, Stomachion, and Measurement of a Circle; in which he detailed his findings of the exact measurement of PI.

            Archimedesí love and dedication to his work may have ultimately cost him his life. In an ironic and tragic turn of events, Archimedes was killed by a Roman soldier. When the soldier approached Archimedes, who was focused on a drawing of circles, Archimedes said ìNoli turbane circulos meos!í This means, ëDo not disturb my circles!í Archimedes was working on a mathematical problem and was so absorbed in it that he became annoyed with the soldier who stepped onto the circles that he was drawingî (HyperHistory.com).  In actuality, the soldier had only come to bring Archimedes to his General. The General was interested in meeting Archimedes because of his reputation for his genius inventions. But, Archimedes offended the soldier with his statement and the soldier killed him. Subsequently, the soldierís General had him executed when he heard that he had killed Archimedes.

            Archimedes clearly made some of the most important contributions to the fields of mathematics and science.  He changed the direction of mathematics as we know it, giving way to the development of calculus, hydrostatics, pycnometry and he furthered research in the field of geometric functions. He came up with many basic principles of physics and invented several revolutionary machines and devices that have continued to be of use to the modern world. Although Archimedes never held an official public office, his lifeís work was dedicated to defending his homeland of Syracuse, and to better the lives of those around him with his mathematical discoveries. Throughout history he has acquired many nicknames; ìThe Father of Integral Calculus,î ìThe Great Geometer,î and ìThe Wise One.î Perhaps the most fitting of these nicknames was ìThe Master.î For not only was he a brilliant man, but his imagination allowed him to push the laws of science with his creative inventions. He became a master of all things mathematical and scientific and is considered to be one of the last great Greek mathematicians.  He found joy when he was deep in thought; left in peace to contemplate pure mathematics.  He devoted his life to his work; not for the fame which has lasted for centuries, but purely for the satisfaction of learning and advancing his own mind. The human race was just fortunate to be learning, and advancing, right along with him.

           

 

Works Cited Sheet

 

Text/Video/News References

 

1. Berlinghoff, William. Grant, Kerry. Skrien, Dale. A Mathematics Sampler: Topics for the Liberal Arts. Maryland: Ardsley House Publishers, 2001.

 

2. Funk & Wagnalls New Encyclopedia. U.S. Funk & Wagnalls Corp, 1993.

 

3. PBS. NOVA Infinite Secrets Series. Archimedes reference webpage. Date Accessed 11/28/03. http://www.pbs.org/wgbh/nova/archimedes/

 

Electronic/Internet References

 

1. University of St. Andrews. The MacTutor History of Mathematics Archive website. Date Accessed 11/22/03. http://www-groups.dcs.st-and.ac.uk/~history/

 

2. University of St. Andrews. Archimedes of Syracuse webpage. Date Accessed 11/18/03. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Archimedes.html

 

3. MERLOT website. Date Accessed 11/18/03. http://www.merlot.org/artifact/BrowseArtifacts.po?catcode=251&browsecat=124

 

4. School for Champions website. Date Accessed 11/22/03.

www.school-for-champions.com/biographies/archimedes.htm

 

5. What You Need To Know About Inventors webpage. Date Accessed 11/22/03. http://inventors.about.com/library/inventors/blarchimedes.htm

 

6. HyperHistory website. Date Accessed 11/22/03. http://www.hyperhistory.net/apwh/bios/b2archimedes_p1ab.htm

 

7. Technology Museum of Thessaloniki website. Date Accessed 11/23/03. http://www.tmth.edu.gr/en/aet/1/13.html

 

8. Enchanted Learning Website. Date Accessed 11/25/03. http://www.enchantedlearning.com/inventors/

 

9. Texas A&M University. Department of Mathematics, Don Allen website. Date Accessed 11/28/03. http://www.math.tamu.edu/~don.allen/history/archimed/archimed.html

 

10. Drexel University. Department of Mathematics, The Golden Crown webpage. Date Accessed 11/28/03. http://www.mcs.drexel.edu/~crorres/Archimedes/Crown/CrownIntro.html

 

11. Saeta, Peter. Physics website, Projectile Motion webpage. Date Accessed 11/28/03. http://kossi.physics.hmc.edu/Courses/p23a/Experiments/Projectile.html

 

12. Integrated Publishing. Mathematics website. Date Accessed 11/28/03. http://www.tpub.com/content/math/

 

13. American Mathematical Society website. Date Accessed 11/29/03. http://www.ams.org/