The Man The Methods of Archimedes feature The Message Introduction Amrutha Manjunath The Sand Reckoner Elsewhere in this issue is a review of by Gillian Bradshaw. That review and this article are dedicated to one of the most celebrated mathematicians in the world. Archimedes is perhaps most famous for the discovery of the Archimedes Principle and the invention of levers, pulleys, pumps, military innovations (like the siege engines) and the Archimedean Screw. His mathematical contributions include approximations of and √3 accurate to several decimal places, proof of the quadrature of the parabola, formula for the area of a circle, and formulae of surface areas and volumes of several solid shapes. In this article, I have focused on two techniques (called Archimedes’ Methods) by which he arrived at the formula of the volume of a sphere. In October 1998, a French family in New York put a thousand-year-old manuscript up for public auction. This manuscript, which the family had acquired in the 1920s, turned out to be a lost Archimedean palimpsest. Byzantine monks in the 13th century had washed the original mathematical text and reused the parchment for Christian liturgical writings. In the early 20th century, Johan Heiberg had studied the same manuscript at Constantinople (present-day Istanbul) and Keywords: Archimedes, volume, cylinder, cone, sphere, method of exhaustion, equilibrium, lever 7 At Right Angles | Vol. 4, No. 3, November 2015 Vol. 4, No. 3, November 2015 | At Right Angles 7 The Method of Equilibrium identi�iedidenti�ied it for theitidenti�ied for �irst the time �irst it for as time work the as �irst work by time by as work by As theAs number the number ofAs cylinders the of numbercylinders increases, of increases, cylinders and the and increases, the and the reveal how closely interlinked the disciplines Archimedes.Archimedes. It disappearedArchimedes. It disappeared for It several disappeared for several years years for several yearsheightheight of each of cylindereachheight cylinder ofcorrespondingly each correspondingly cylinder correspondingly Now I will discuss the second technique by which really are. duringduring the aftermath the aftermathduring of the of Greco-Turkishaftermath the Greco-Turkish of the War, Greco-Turkish and War, and decreases, War,decreases, and the sum thedecreases, of sum volumes of thevolumes ofsum the of of cylinders volumes the cylinders is of a the is cylinders a is a Archimedes arrived at the same result for the To �ind the volume of a sphere by the Method of resurfacedresurfaced in the inresurfaced possession the possession in of the the possessionof French the French of the Frenchclosercloser and closer and closer approximation approximationand closer to approximation the to volume the volume of to the of volume of volume of a sphere. This technique, known as the Equilibrium, it helps to think of the solid as cut up businessmanbusinessman whosebusinessman whose descendants descendants whose put descendants it put up for it up for put it upthe for hemisphere.the hemisphere.the Therefore, hemisphere. Therefore, as approaches asTherefore, approaches as approaches Method of Equilibrium, was found in the lost into a large number of very thin strips hung end auction.auction. From From 1999auction. 1999 to 2008, From to 2008, the 1999 manuscript the to manuscript 2008, was the manuscript was in�inity, wasin�inity, the sum thein�inity, of sum the of volumes the sumvolumes of of the the of cylinders volumes the cylinders of the cylinders palimpsest. It sheds light on a very uniquely to end on an imaginary lever. 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Since volume is proportional to mass, collaborationcollaboration withcollaboration scientists with scientists at with Rochester at scientists Rochester Institute at Institute Rochester Institute � � � � �� � � � � � � = ���lim =����lim �+ = ���lim+�+ +++···+ +···++ �+···+� � argument that resonates with the modern notion moment of the solid can be de�ined as the product of Technologyof Technology andof Stanford Technology and Stanford University. and University. Stanford Many ManyUniversity. Many � �� �� �� �� � � of integral calculus. of its volume and lever length (the distance from ArchimedeanArchimedean textsArchimedean weretexts wererecovered recovered texts from were thisfrom recovered this from this � �� �� �� �� � � = ���lim=���lim( +(= ���+lim +···++( +···++ +. +···+. the point about which the shapes are hung to the palimpsest,palimpsest, of whichpalimpsest, of which the work the of work on which Methods on the Methods work is on is Methods is There has been considerable debate among centroid of the volume). especiallyespecially interesting interestingespecially to many interestingto many mathematicians. mathematicians. to many mathematicians. � � � �� � mathematicians about which Method of � � � TheThe Method Method ofThe Exhaustion of Method Exhaustion of Exhaustion SubstituteSubstitute =Substitute−= (− (for= for≤− (≤≤:≤for: ≤≤: Archimedes is the superior one. Archimedes Figure 3 is a cross-sectional view along the � � � � � � �� � � � conceptualized notions of limits and integration equator of the sphere. Here . ( +2( +3+2 +···++3( +···++2 +3 +···+ well before calculus emerged as a powerful Consider the cylinder and cone of revolution The MethodThe Method of ExhaustionThe of Exhaustion Method is ofa well-known Exhaustion is a well-known is a well-known = ���lim =���lim� −=����lim− ��− � � � � � mathematical tool, so both methods contain ideas obtained by rotating rectangle and triangle techniquetechnique using usingtechnique which which the usingarea the of area which a �igure of the a �igure can area be can of a be �igure can be � � � � � � �� � �� � � � much ahead of their time. Historians of about the axis. Suppose thin vertical foundfound by visualizing by visualizingfound it toby be visualizing it to composed be composed it toof be of composed of ==− −���lim=���lim��−�+2� ���lim+3+2 +···++3� � +···++2 �.+3 �.+···+ �. mathematics such as Howard Eves argue that the slices of thickness Δ are cut from the three solids constituentconstituent polygonsconstituent polygons that converge that polygons converge to that the to area converge the of area to of the area of Method of Exhaustion is “sterile” because its at distance from . The approximate volumes of the containingthe containing shape.the containing shape. 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Among Among otherlimits. other results, Among results, Archimedes other Archimedes results, used Archimedes usedthe the used the Equilibrium is unique for Archimedes’ use of The equation of the circular MethodMethod of Exhaustion of ExhaustionMethod to of compute Exhaustion to compute the volume to the compute volume of a the of a volume of a cross-section� � of the� sphere� is � � � � �(+�((2++(2(+ +(2 + mechanics to prove a purely mathematical result. sphere.sphere. 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However, layeringlayering of cylinders oflayering cylinders inscribed of inscribed cylinders within within inscribed the solid. the solid. within the solid.==− −= == . − . = . The volume of revolution of the slice of� Let theLet radius the radius ofLet the of the hemisphere the radius hemisphere of bethe hemisphere, andbe , radii and ofradii be of, and radii of 3 3 3 3 3 3 in the words of E.T. Bell, “To a modern all is fair in Cylinder:cone with thickness Δ and height is Δ. each cylindereach cylinder beeach�, be�, cylinder�,,…,�,��,…,. be If there�,. If�, there are�,…, are�.