Archimedes’ Determination ofCircular Areapresentation byAustin Cornett, Erin Loos,Ben SchadlerMA 330 - History of MathematicsArchimedes of Syracuse! Born 287 B.C. at Syracuse! Father was an astronomer! Developed life-long interest in the heavens! Most of what we know about Archimedescomes from Plutarch’s Life of Marcellus andArchimedes writings to the scholars atAlexandriaArchimedes (Cont’d)! Spent time in Egypt as boy! Studied at the great Library ofAlexandria! Trained in the Euclidean tradition! Created “Archimedean screw”Archimedes (Cont’d)! Archimedes wrote a treatise entitled OnFloating Bodies, which discussed theprinciples of hydrostatics! Archimedes invented a water pump, knownas the Archimedean Screw, which is stillused today! Archimedes was instrumental in thedevelopment of pulleys, levers, and opticsMarcellus’ Siege! Romans invaded Syracuse, led by theGeneral Marcellus! In defense of his homeland,Archimedes developed weapons toward off the RomansMarcellus’ (Cont’d)! Archimedes prepared the city forattacks during both day and night! The Syracusans dropped their guardduring feast to Diana! Opportunistic Romans seized theirchance and invaded the city! The death of Archimedes broughtgreat sorrow to MarcellusThe Death of Archimedes! …as fate would have it, intent upon working outsome problem by a diagram, and having fixed hismind alike and his eyes upon the subject of hisspeculation, [Archimedes] never noticed theincursion of the Romans, nor that the city wastaken. In this transport of study and contemplation,a soldier, unexpectedly coming up to him,commanded him to follow to Marcellus; which hedeclining to do before he had worked out hisproblem to a demonstration, the soldier, enraged,drew his sword and ran him through.PlutarchAncients Knowledge of CircularArea Pre Archimedes! The ancients knew the ratio of C over Dwas equal to the value !! Proposition 12.2 of Euclid states the ratio ofcircular area to D2 is constant! The Area of a regular polygon is "hQ! They knew that an inscribed polygon’s areawould be less than that of a circleregardless of how many sides it had! !Measurement of a Circle! Proposition I" The area of any circle is equal to a right-angled triangle in which one of the sidesabout the right angle is equal to theradius, and the other to thecircumference of the circleMeasurement (cont’d)! Proof" Archimedes began with two figures:! a circle with a center O, radius r, and circumference C! a right triangle with base of length C and height of r" the area of the Circle being equal to A" the area of the Triangle equal to T" To prove A=T, Archimedes used a doublereductio ad absurdumMeasurement (cont’d)! Case 1" Suppose A>T then A-T is positive" Archimedes knew by inscribing a squarewithin his circle and repeatedly bisectingit he could create a regular polygon witharea Z with A-Z < A-T" Adding Z+T-A to both sides of A-Z < A-T! T < ZCase 1 (cont’d)" But this is an inscribed polygon thereforethe perimeter of the polygon Q is lessthan C (the circumference of the circle)and it’s apothem is less than the circle’sradius" Hence Z = "hQ < "rC = T (contradiction)Measurement (cont’d)! Case 2" Suppose A < T then T-A > 0" Circumscribe about the circle a regularpolygon who’s area exceeds the circle byno more than T-A" With area of circumscribed polygon = Z! Z-A < T-A" Adding A to both sides results in Z < TCase 2 (cont’d)" But the apothem of the polygon equals rwhile the polygon’s perimeter Q exceedsthe circle’s circumference" Thus Z = "hQ > "rC = T (contradiction)" Therefore A must equal T" Q.E.D.Results of Proposition I! Archimedes had related a circles areanot to that of another circle as Eucliddid, but to its own circumference andradius! Remembering that C = !D = 2!r, werephrase his theorem as" A = "rC = "r(2!r) = !r2Archimedes’ Other Workspresentation byAustin Cornett, Erin Loos,Ben SchadlerMA 330 - History of MathematicsArhimedes’ Method of Discovery! Presented his method of discoverybefore each proof! The “Method” is a treatise of theresults of his methods! Discovered in 1899 unexpectedly" Parts of it had been washed out! Sold for two million! rapMeasurement of a Circle! Proposition 3" The ratio of the circumference of any toits diameter is less than but greaterthan .! 31071! ! 317Proposition 3 (cont’d)! Used inscribed and circumscribedpolygons starting with a hexagon! Archimedes bisected sides ofpolygons until had 96-gon! At each stage he had to approximatesquare roots" Very difficult to do back thenA Chronology of !! Ca. 240 B.C. - Archimedes! Ca. A.D. 150 - Ptolemy! Ca. 480 - Tsu Ch’ung-chih! Ca. 530 - Aryabhata! Ca. 1150 - Bhaskara! 1429 - Al-Kashi! (cont’d)! 1579 - Francois Viete! 1585 - Adriaen Anthoniszoon! 1593 - Adriaen von Roomen! 1610 - Ludolph van Ceulen! 1621 - Willebrord Snell! 1630 - Grienberger! (cont’d)! 1650 - John Wallis! 1671 - James Gregory! 1699 - Abraham Sharp! 1706 - John Machin! 1719 - De Lagny! 1737 - William Oughtred, Isaac Barrow, David Gregory! (cont’d)! 1754 - Jean Etienne Montucla! 1755 - French Academy of Sciences! 1767 - Johann Heinrich! 1777 - Comte de Buffon! 1794 - Adrien-Marie Legendre! 1841 - William Rutherford! 1844 - Zacharias Dase! 1853 - Rutherford! 1873 - William Shanks! 1882 - F. Lindemann! 1906 - A. C. Orr! 1948 - D. F. Ferguson! (cont’d)! 1949 - ENIAC! 1959 - Francois Genuys! 1961 - Wrench and Daniel Shanks! 1965 - ENIAC! 1966 - M. Jean Guilloud! 1973 - Guilloud again! (cont’d)! 1981 - Kazunori Miyoshi and Kazuhika Nakayama! 1986 - D. H. Bailey! (cont’d)On the Sphere and the Cylinder! Considered Archimedes masterpiece! Achieved for 3D bodies whatMeasurement of a Circle had done for2D bodies! Proposition 13" The surface of any right circular cylinderexcluding the bases is equal to a circlewhose radius is a mean proportionalbetween the side of the cylinder and thediameter of the base.On the Sphere and the CylinderProposition 13 (cont’d)! In modern terms lateral surface ofcylinder (height h, radius r) is equal toarea of a circle w/ radius x! hx=x2r! Proposition 33" The surface of any sphere is equal tofour times the greatest circle in itOn the Sphere and the CylinderProposition 33 (cont’d)! Archimedes proved using double reductioad absurdam! Can restate proof as the surface area of asphere is equal to 4!r2! There’s nothing intuitive regarding aboutthis result! Archimedes states he was only luckyenough to glimpse at these internal