Heather Hatlo

Professor Walton

Math G

December 1, 2003

TheMathematics of Archimedes

Archimedesis considered by most historians to be one of the greatest mathematicians ofall time. He is also thought of as one of the greatest scientists of all time, knownfor many important inventions. His fields of study included; hydrostatics,buoyancy, static machines, and pycnometry (the measurement of the volume ordensity of an object). He has been nicknamed the “father of integral calculus,”for the tremendous impact he had on the study of integration and calculus. Archimedeswas instrumental in assisting the armies of his homeland to defend themselvesagainst the Romans. His war machines helped avoid invasion for many years andin turn earned him praise throughout the region. While he was an extremelyfamous inventor in his day, he took much more pride and pleasure in puremathematics. He would get so engrossed in solving mathematical equations thatit was common for him to skip meals and neglect to bathe for days at a time. Itwasn’t until after his death that he really began to receive recognition forhis mathematic contributions. Even without recognition during his life,Archimedes preferred to spend his days theorizing rather than inventinganything tangible.

Archimedeswas born in 287 B.C. in Syracuse, Sicily. His father, Phidias, was anastronomer. This had an important impact on Archimedes’ future studies. He wasfascinated by his father’s work and began learning about Astronomy at a veryyoung age. Having educated parents gave Archimedes an edge over his peers. Hisparents 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 veryinquiring mind and delved into many fields of study when he went away toschool. Another advantage Archimedes was born into was his relationship withKing Heiron II. Although there is not specific evidence that this relationshipcontributed to his rise as an important figure of his time, it is very likelythat in a monarchal society it did have some affect on Archimedes’ stature inthe community. While we do not know if this relationship was familial or justfriendly, we do know it was an important relationship to Archimedes because ofbook dedications he made to members of the royal family, and prefaces he wrotein his book.

Thegreatest contribution Archimedes made to the mathematical world was his work inthe study of integration. He was fascinated with discovering volumes and theareas of many objects. He found ways to calculate the areas of difficult tomeasure shapes, like parabolas and ellipses. To discover these measurements,Archimedes broke up images into an infinite number of rectangles and then addedthe areas of the sections together. An example of this can be seen in the imageon the right. This is known as the method of exhaustion; “a technique ofapproximating the area of a region with an ever increasing number of polygons,with the approximations improving at each step and the exact area beingattained after an infinite number of these steps” (Guide to History ofCalculus). Specifically, for solving the quadrature of a parabola, he firstapproximated the area with a large number of triangles and then used a “doublereductio ad absurdum argument” (Guide to History of Calculus) to prove hisresult. Archimedes’ work in integration would provide the foundation for thedevelopment of calculus later in history.

Integrationwasn’t the only area of mathematics that Archimedes made huge advances in. Archimedeswas particularly fascinated with geometric applications in astronomy, which isnot surprising considering his father’s profession. He was the first person tocalculate the length of a year, after building a machine to measure the anglesof the rising sun. He made many other significant contributions to the field ofgeometry; among the most important was his discovery of a very accurateapproximation of PI. He determined it to be somewhere between 3.1408 and3.1428. He computed this as follows:

With respect to a circle of radius r, let

Further, let denotethe regular inscribed polygons,similarly, forthe circumscribed polygons. The following formula gives the relation betweenthe perimeters and areas of these polygons.

Using n-gons up to96 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 acircle. Then he determined “the length of the perimeter of a polygoninscribed within a circle (which is less than the circumference of the circle)and the perimeter of a polygon circumscribed outside a circle (which is greaterthan the circumference)” (PBS, NOVA series).  Although there had been many close approximations of PIprior to Archimedes’ discovery, this was the first time it had been calculatedtheoretically rather than actually physically measured. It was also the mostaccurate measurement at the time.  Evento this day, with super computers measuring PI out to one trillion decimalplaces, Archimedes’ calculation can be used for most practical applications.However important the calculation of PI has been to the world, Archimedesconsidered a different calculation to be his most important in geometry. He provedthat the volume of a sphere is two-thirds the volume of a cylinder thatcircumsribes the sphere. He felt this was so important in fact, that herequested an illustration of this be inscribed on his tombstone.

            Withall of the time and effort Archimedes spent on solving math equations andanalyzing complex problems, he soon found he could solve a number of practicalproblems by putting his mathematical skills to good use. He was often called onby the King to solve problems for the monarchy. One of the issues he resolvedfor King Heiron II, is known as the Gold Crown problem. The King had hired agoldsmith to make him a crown of pure gold. However, upon delivery of thecrown, the King suspected some silver had been used. He called on Archimedes tohelp prove this. After spending some time thinking this issue over, Archimedesdiscovered a simple way to determine the crown’s composition while he soaked in abathtub one afternoon. He realized that the displaced water in the tub wasproportional to the weight of the object submerged in the water. Thisobservation is now known as Archimedes’ Principle, and it opened up the door tothe study of hydrostatics and buoyancy. To prove the crown was indeed made withsome silver, Archimedes took an amount of gold that should have been equal tothe amount in the crown. Placing them both in a tub of water, it was clear thatthe crown did not displace as much water, and this proved the fraudulent workof the goldsmith. The King was so pleased with Archimedes, that he requested hisassistance in the fight against the Romans. Archimedes invented a number ofmachines and devices that could be used to defend Sicily during the First andSecond Punic Wars.

            Theweapons that Archimedes invented for war fall into three main categories: a)cranes or claws, b) catapults of every size and description, and c) mirrors thatfocused sunlight on ships and set them alight.  Probably themost talked about of these inventions, was his system of mirrors that could befocused on enemy ships in the harbor. These mirrors would then reflect rays ofsunlight directly on the ships and cause them to catch fire and burn. Thisinvention became legendary, and for a long time scholars  debated whether or not such an inventioncould have been possible in the time of Archimedes.  For centuries it was believed that Archimedes’ mirrors wereonly a myth. That was until an engineer proved it was possible in 1973. Using70 copper plated glass lenses, Ioannis Sakkas conducted an experiment on theisland of Salamina. He placed the lenses in a circle and focused the suns rayson a small boat drifting 55 feet out in the water. The boat was crafted in asimilar fashion to the Roman ships of Archimedes’ day. Within moments the boatwas engulfed in flames, proving Archimedes’ invention had worked. Otherimportant inventions used in the punic wars include; cranes thatlifted enemy ships out of the water and dashed them against the rocks, andcatapults that hurled bolts and stones varying distances.

            Thesuccessful use of Archimedes’ catapults was due to his great understanding ofhow levers and pulleys operate. He originally created a system of levers toassist the King with more traditional work in the kingdom, but they proved veryuseful as weapons of defense. His first description of the lever was around260BC.  He applied thisunderstanding of levers to create remarkable catapults that could tossquarter-ton stones at Roman ships in the harbor. Archimedes is quoted as saying;“Give me a lever long enough, and aplace to stand, and I will move the world” (Archimedes, 230BC). This statementhas become an inspiration to millions of inventors, engineers and scientistsalike.

            Alongwith other uses for his system of levers and pulleys, many of Archimedes’inventions were not used only for war. He enjoyed solving practical problemswith his inventions, like the keeping of time for example. During hislife,  the Greeks had a very goodsystem of tracking the hours in the day, but once the sun went down, and theycould 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. Archimedessolved this problem with his creation of a more advanced water clock. His new deviceallowed for accurate calculations of time during the day and night, with amargin of error of only two minutes. While water clocks had existed for sometime, Archimedes added gears and also showed the movement of the planets andthe orbiting moon. Another signifigant invention was Archimedes’ Screw. He inventedthis device around 269BC, and it was used to transport water with only a smallamount 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. Oneform of this screw consisted of a circular pipe enclosing a helix, and theninclined at a forty-five degree angle to the horizontal with the lower enddipped into the water. When you rotate the device, the water rises into thepipe. This invention was so important, that it is still used in the present dayin countries around the world for irrigation of crops and in water treatmentcenters. The picture on the left is “one of eight 12-ft.-diameter Archimedes’screws used to handle rainstorm runoff in Texas City, Texas. Each screw isdriven by a 750-hp diesel engine and can pump up to 125,000 gallons per minute”(Drexel University).

            Weare fortunate to know so much about Archimedes’ inventions and mathematicalcalculations because of the many literary works he left behind. These bookshave proved to be invaluable in the study of mathematics. Some of theseimportant works include The Quadrature of the Parabola; in whichArchimedes calculated the area of a segment of a parabola, as was discussedearlier in this paper. Archimedes wrote two volumes of On Floating Bodies;books that detailed the law of equilibrium of fluids and proved that wateraround a center of gravity will take a spherical form.  Another important contribution camefrom the book The Sand Reckoner. It was in this that “Archimedesproposed a number system capable of expressing numbers up to 8x10¹⁶in modern notations. He argued in this work that this number is large enough tocount 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 togive the dimensions of the universe to be able to count the number of grains ofsand which it could contain” (University of St. Andrews).  The book On the Sphere and Cylindercontained the works that Archimedes was most proud of; the discovery that thearea and volume of a sphere are in the same relationship as the area and volumeof a circumscribed cylinder.

            Hismost recent work, The Method, was only discovered in the early nineteenhundreds. It was discovered in a larger set of documents, known as TheArchimedes Palimpset. An image of these documents is shown on the pageabove. The Method has been an important insight into the mind ofArchimedes. It has helped give mathematicians and scientists an understandingof 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 oneanother, and calculated the volumes and areas with those measurements. AlthoughArchimedes did not rely on the information in The Method as proof of hiscalculations; (instead he preferred to publish work done with the method ofexhaustion), this book has proved to be great research material for thosestudying Archimedes’ work. Other well known literary works of Archimedesinclude; On Plane Equilibriums, On Spirals, On Conoids andSpheroids, Stomachion, and Measurement of a Circle; in whichhe detailed his findings of the exact measurement of PI.

            Archimedes’love and dedication to his work may have ultimately cost him his life. In anironic and tragic turn of events, Archimedes was killed by a Roman soldier.When the soldier approached Archimedes, who was focused on a drawing ofcircles, Archimedes said “Noli turbane circulos meos!’ This means, ‘Do notdisturb my circles!’ Archimedes was working on a mathematical problem and wasso absorbed in it that he became annoyed with the soldier who stepped onto thecircles that he was drawing” (HyperHistory.com).  In actuality, the soldier had only come to bring Archimedesto his General. The General was interested in meeting Archimedes because of hisreputation for his genius inventions. But, Archimedes offended the soldier withhis statement and the soldier killed him. Subsequently, the soldier’s Generalhad him executed when he heard that he had killed Archimedes.

            Archimedesclearly made some of the most important contributions to the fields ofmathematics and science.  Hechanged the direction of mathematics as we know it, giving way to thedevelopment of calculus, hydrostatics, pycnometry and he furthered research inthe field of geometric functions. He came up with many basic principles of physicsand invented several revolutionary machines and devices that have continued tobe of use to the modern world. Although Archimedes never held an officialpublic office, his life’s work was dedicated to defending his homeland ofSyracuse, and to better the lives of those around him with his mathematicaldiscoveries. Throughout history he has acquired many nicknames; “The Father ofIntegral Calculus,” “The Great Geometer,” and “The Wise One.” Perhaps the mostfitting of these nicknames was “The Master.” For not only was he a brilliantman, but his imagination allowed him to push the laws of science with hiscreative inventions. He became a master of all things mathematical andscientific and is considered to be one of the last great Greek mathematicians.  He found joy when he was deep inthought; left in peace to contemplate pure mathematics.  He devoted his life to his work; not forthe fame which has lasted for centuries, but purely for the satisfaction oflearning and advancing his own mind. The human race was just fortunate to belearning, 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. NOVAInfinite Secrets Series. Archimedes reference webpage. Date Accessed11/28/03. http://www.pbs.org/wgbh/nova/archimedes/

Electronic/InternetReferences

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

2.University of St. Andrews. Archimedes of Syracuse webpage. Date Accessed11/18/03. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Archimedes.html

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

4. Schoolfor Champions website. Date Accessed 11/22/03.

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

5. WhatYou 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. TechnologyMuseum of Thessaloniki website. Date Accessed 11/23/03. http://www.tmth.edu.gr/en/aet/1/13.html

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

9. TexasA&M University. Department of Mathematics, Don Allen website. DateAccessed 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 Accessed11/28/03. 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. AmericanMathematical Society website. Date Accessed 11/29/03. http://www.ams.org/