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 𝜋𝜋𝜋𝜋 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 𝑛𝑛𝑛𝑛 approaches asTherefore,𝑛𝑛𝑛𝑛 approaches as 𝑛𝑛𝑛𝑛 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. This proof compares subjectsubject to extensive to extensivesubject imaging to imaging extensive study study and imaging conservation and conservation study and conservationequalsequals the volume the volumeequals𝑉𝑉𝑉𝑉 of the𝑉𝑉𝑉𝑉 of volumehemisphere. the hemisphere.𝑉𝑉𝑉𝑉 of the That hemisphere. is,That is, That is, Archimedean way of thinking about surface areas the moments of two solids when placed on the at theat Walters the Walters Artat Museum the Art Walters Museum in Baltimore Art in Museum Baltimore in in in Baltimore in � � � � � � � � � � � � and volumes of solid shapes, and employs an lever. Since volume is proportional to mass, collaborationcollaboration withcollaboration scientists with scientists at with Rochester at scientists Rochester Institute at Institute Rochester Institute � 𝑟𝑟𝑟𝑟 � 𝑟𝑟𝑟𝑟� 𝑟𝑟𝑟𝑟 � �𝑟𝑟𝑟𝑟�𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟 � �𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟 � �𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟 � 𝑟𝑟𝑟𝑟 � 𝑟𝑟𝑟𝑟 𝑉𝑉𝑉𝑉 = ���lim𝑉𝑉𝑉𝑉 =�𝜋𝜋𝜋𝜋���lim𝜋𝜋𝜋𝜋 �𝜋𝜋𝜋𝜋𝑉𝑉𝑉𝑉+𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋 = 𝑟𝑟𝑟𝑟���lim+𝜋𝜋𝜋𝜋�𝜋𝜋𝜋𝜋+𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟 𝜋𝜋𝜋𝜋 𝑟𝑟𝑟𝑟 +𝜋𝜋𝜋𝜋+𝜋𝜋𝜋𝜋+···+𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟 +···+𝜋𝜋𝜋𝜋+𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟 �𝑟𝑟𝑟𝑟+···+𝜋𝜋𝜋𝜋� 𝑟𝑟𝑟𝑟 � 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 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 � �� �� �� �� � � 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 𝑟𝑟𝑟𝑟 � 𝑟𝑟𝑟𝑟 �� �� 𝑟𝑟𝑟𝑟 �� �� � � = ���lim=𝜋𝜋𝜋𝜋���lim(𝑟𝑟𝑟𝑟𝜋𝜋𝜋𝜋 +𝑟𝑟𝑟𝑟(𝑟𝑟𝑟𝑟= ���+𝑟𝑟𝑟𝑟lim 𝜋𝜋𝜋𝜋+···+𝑟𝑟𝑟𝑟+𝑟𝑟𝑟𝑟(𝑟𝑟𝑟𝑟 +···+𝑟𝑟𝑟𝑟+𝑟𝑟𝑟𝑟 𝑟𝑟+𝑟𝑟𝑟𝑟. 𝑟𝑟+···+𝑟𝑟𝑟𝑟. 𝑟𝑟. 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 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 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 SubstituteSubstitute𝑟𝑟𝑟𝑟 =𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟Substitute−=𝑟𝑟𝑟𝑟 (𝑗𝑗𝑗𝑗𝑟𝑟𝑟𝑟𝑟𝑟−𝑟𝑟𝑟𝑟 (𝑗𝑗𝑗𝑗𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟for=𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑛𝑛 for≤𝑗𝑗𝑗𝑗− (𝑗𝑗𝑗𝑗𝑛𝑛≤𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟𝑟𝑟≤𝑗𝑗𝑗𝑗𝑟𝑟𝑟𝑟:𝑟𝑟≤𝑛𝑛𝑛𝑛for: 𝑛𝑛 ≤𝑗𝑗𝑗𝑗≤𝑛𝑛𝑛𝑛: 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 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴. 𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟 (𝑛𝑛 +2(𝑛𝑛𝑟𝑟𝑟𝑟 +3+2 +···+𝑛𝑛𝑛𝑛+3(𝑛𝑛 +···+𝑛𝑛𝑛𝑛+2 𝑟𝑟+3 𝑟𝑟+···+𝑛𝑛𝑛𝑛 𝑟𝑟 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-known𝑉𝑉𝑉𝑉 = ���lim𝑉𝑉𝑉𝑉 =𝜋𝜋𝜋𝜋���lim�𝑛𝑛𝑛𝑛𝜋𝜋𝜋𝜋 𝑉𝑉𝑉𝑉�=�𝑛𝑛𝑛𝑛���lim� 𝜋𝜋𝜋𝜋 �𝑛𝑛𝑛𝑛�� � � � � � 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 mathematical tool, so both methods contain ideas obtained by rotating rectangle 𝐴𝐴𝐴𝐴𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝐴𝐴𝐴𝐴 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 � � � � 𝑛𝑛 � �𝑛𝑛 �� �𝑛𝑛 �� � � � much ahead of their time. Historians of 𝐴𝐴𝐴𝐴𝑂𝑂𝑂𝑂𝐴𝐴𝐴𝐴 about the 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 axis. Suppose thin vertical foundfound by visualizing by visualizingfound it byto be visualizing it to composed be composed it toof be of composed of =𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟=𝜋𝜋𝜋𝜋−𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 −𝜋𝜋𝜋𝜋���lim𝑟𝑟𝑟𝑟=𝜋𝜋𝜋𝜋���lim�𝑟𝑟𝑟𝑟�𝑛𝑛−𝜋𝜋𝜋𝜋�+2�𝑛𝑛𝑟𝑟𝑟𝑟 ���lim+3+2 +···+𝑛𝑛𝑛𝑛+3� �𝑛𝑛 +···+𝑛𝑛𝑛𝑛+2 �.+3 �.+···+𝑛𝑛𝑛𝑛 �. 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 mathematics such as Howard Eves argue that the slices of thickness Δ𝑥𝑥𝑥𝑥 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 𝑥𝑥𝑥𝑥 from 𝐴𝐴𝐴𝐴. The approximate volumes of the containingthe containing shape.the containing shape. It is considered It is shape. considered to It isbe considered to the be the to be the Now useNow the use formula theNow formula use the formula elegance is apparent only if the result is already theSphere: sections of each solid are deduced to be: ancient-Greekancient-Greek equivalentancient-Greek equivalent of the equivalentof modern the modern notion of the notion of modern of notion� � of � �� � �� � � � 𝑛𝑛 +2𝑛𝑛 +3+2 +···+𝑛𝑛𝑛𝑛+3𝑛𝑛 +···+𝑛𝑛𝑛𝑛+2 +3= � 𝑛𝑛𝑛𝑛+···+𝑛𝑛𝑛𝑛=(𝑛𝑛𝑛𝑛 �+𝑛𝑛𝑛𝑛𝑛𝑛(𝑛𝑛𝑛𝑛𝑟𝑟𝑟+𝑛𝑛𝑛𝑛𝑛𝑛=+𝑟𝑟𝑟�𝑛𝑛𝑛𝑛𝑛𝑛𝑟𝑟:(+𝑛𝑛𝑛𝑛 +𝑛𝑛𝑟𝑟𝑛𝑛: 𝑟𝑟𝑟𝑛𝑛𝑛𝑛 + 𝑛𝑛𝑟𝑟: known. While this is debatable, the Method of limits.limits. 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 � � � � 𝑛𝑛�𝑛𝑛𝑛𝑛(𝑛𝑛𝑛𝑛𝑛𝑛+𝑛𝑛𝑛𝑛�𝑛𝑛(𝑛𝑛𝑛𝑛𝑟𝑟𝑟+𝑛𝑛𝑛𝑛𝑛𝑛+𝑟𝑟𝑛𝑛𝑟𝑛𝑛𝑛𝑛𝑛𝑛𝑟𝑟(+𝑛𝑛𝑛𝑛 +𝑛𝑛𝑟𝑟𝑛𝑛𝑟𝑟𝑟𝑛𝑛𝑛𝑛 + 𝑛𝑛𝑟𝑟 mechanics to prove a purely mathematical result. sphere.sphere. I have I discussedhavesphere. discussed I this have method this discussed method below this below using method using below𝑉𝑉𝑉𝑉 using= 𝜋𝜋𝜋𝜋𝑉𝑉𝑉𝑉𝜋𝜋𝜋𝜋=−𝜋𝜋𝜋𝜋 𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟 −𝜋𝜋𝜋𝜋���lim𝑉𝑉𝑉𝑉𝑟𝑟𝑟𝑟=���� 𝜋𝜋𝜋𝜋lim𝜋𝜋𝜋𝜋� −𝜋𝜋𝜋𝜋� �𝑟𝑟𝑟𝑟 ���lim � � � � � Archimedes himself is said to have preferred the (𝑥𝑥𝑥𝑥 𝑥𝑥 𝐴𝐴𝐴𝐴𝑥𝑥 +𝑦𝑦𝑦𝑦 𝐴𝐴𝐴𝐴𝐴𝐴 , i.e., 𝑦𝑦𝑦𝑦 𝐴𝐴 𝑥𝑥𝑥𝑥(𝐴𝐴𝐴𝐴𝐴𝐴 𝑥𝑥. 𝑥𝑥𝑥𝑥𝑥𝑥 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 6 𝑛𝑛𝑛𝑛6 6 modernmodern notation. notation.modern notation. � �� � � � Method of Exhaustion because he felt that it was Therefore the volume of revolution of the � � � � � � 𝑛𝑛 𝑛𝑛𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 slice of� sphere with thickness Δ𝑥𝑥𝑥𝑥 and height ConsiderConsider the hemisphere theConsider hemisphere inthe Figure hemisphere in Figure 1. Archimedes 1. in Archimedes Figure 1. Archimedes=𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟=𝜋𝜋𝜋𝜋−𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 −𝜋𝜋𝜋𝜋���lim𝑟𝑟𝑟𝑟=𝜋𝜋𝜋𝜋����lim𝑟𝑟𝑟𝑟� �−𝜋𝜋𝜋𝜋� �𝑟𝑟𝑟𝑟+� ���lim++� ���+� ��+ + �� mathematically more rigorous. Perhaps this was 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛3 32 𝑛𝑛𝑛𝑛62 36 2 6 Cone:𝑦𝑦𝑦𝑦 is 𝜋𝜋𝜋𝜋𝑦𝑦𝑦𝑦 Δ𝑥𝑥𝑥𝑥 = 𝜋𝜋𝜋𝜋𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥� 𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥�𝑥𝑥𝑥𝑥. imaginedimagined the hemisphere theimagined hemisphere tothe be hemisphere to formed be formed by tothe by be the formed by the � � � � � � born out of his innate preference for pure � 𝜋𝜋𝜋𝜋�𝑟𝑟𝑟𝑟 𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟2𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟 �2𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟 2𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟 mathematics to mechanical inventions. However, layeringlayering of cylinders oflayering cylinders inscribed of inscribed cylinders within within inscribed the solid. the solid. within the solid.=𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟=𝜋𝜋𝜋𝜋− 𝑟𝑟𝑟𝑟 −= =𝜋𝜋𝜋𝜋=𝑟𝑟𝑟𝑟 . − . = . The volume of revolution of the slice of� Let theLet radius the radius ofLet the of the hemisphere the radius hemisphere of bethe𝑟𝑟𝑟𝑟 hemisphere, andbe 𝑟𝑟𝑟𝑟, radii and ofradii be 𝑟𝑟𝑟𝑟 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 Δ𝑥𝑥𝑥𝑥 and height 𝑥𝑥𝑥𝑥 is 𝜋𝜋𝜋𝜋𝑥𝑥𝑥𝑥 Δ𝑥𝑥𝑥𝑥. each cylindereach cylinder beeach𝑟𝑟𝑟𝑟�,𝑟𝑟𝑟𝑟 be�,𝑟𝑟𝑟𝑟 cylinder𝑟𝑟𝑟𝑟�,𝑟𝑟𝑟𝑟,…,𝑟𝑟𝑟𝑟�,𝑟𝑟𝑟𝑟��,…,𝑟𝑟𝑟𝑟. be If there𝑟𝑟𝑟𝑟�,𝑟𝑟𝑟𝑟. If�,𝑟𝑟𝑟𝑟 there are�,…,𝑟𝑟𝑟𝑟𝑛𝑛𝑛𝑛 are�. If𝑛𝑛𝑛𝑛 there are 𝑛𝑛𝑛𝑛 love, war, and mathematics.” Maybe the SinceSince𝑉𝑉𝑉𝑉 is half𝑉𝑉𝑉𝑉 is the halfSince required the𝑉𝑉𝑉𝑉 requiredis half volume, the volume, required the volume the volume, volume of the of volume of The volume of revolution of the slice of cylinderscylinders of equal ofcylinders equal height height laid of equal one laid on heightone top on of laid top one oneof one on top of one � � � � � � equilibrium argument is considered more elegant the spherethe sphere with radiusthe with sphere radius𝑟𝑟𝑟𝑟 is given with𝑟𝑟𝑟𝑟 is given radius by � 𝜋𝜋𝜋𝜋 by𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 .is� 𝜋𝜋𝜋𝜋 given𝑟𝑟𝑟𝑟 . by � 𝜋𝜋𝜋𝜋𝑟𝑟𝑟𝑟 . cylinder with thickness Δ𝑥𝑥𝑥𝑥 and height 𝐴𝐴𝐴𝐴𝐴𝐴 is another,another, it follows it followsanother, that the that it heightfollows the height of that each of the cylindereach height cylinder of each cylinder today because there is something enchanting � 𝜋𝜋𝜋𝜋𝐴𝐴𝐴𝐴 Δ𝑥𝑥𝑥𝑥. is 𝑟𝑟𝑟𝑟𝑟𝑟𝑛𝑛𝑛𝑛is. By𝑟𝑟𝑟𝑟𝑟𝑟𝑛𝑛𝑛𝑛 the. By Pythagorean theis 𝑟𝑟𝑟𝑟𝑟𝑟 Pythagorean𝑛𝑛𝑛𝑛. By the Theorem: Pythagorean Theorem: Theorem: A similarA similar argument argumentA similar can be can argument made be made to obtain can to be obtain the made the to obtain the when borders between related disciplines melt to � � � � � � samesame result result if thesame if hemisphere the result hemisphere if isthe thought hemisphere is thought to be to is bethought to be � 𝑟𝑟𝑟𝑟� 𝑟𝑟𝑟𝑟 (2𝑟𝑟𝑟𝑟𝑟𝑟� (2𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 (2𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟� =𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟� −=𝑟𝑟𝑟𝑟� −,𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟� ,𝑟𝑟𝑟𝑟−=𝑟𝑟𝑟𝑟� −� ,� ,𝑟𝑟𝑟𝑟� …,,� − …, � , …,circumscribedcircumscribed bycircumscribed a layering by a layering of cylinders; by of a layeringcylinders; see of see cylinders; see 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 � � � � � Figure�Figure 2. The 2. solid TheFigure solidshape 2. shape isThe in solid fact is in “sandwiched” shapefact “sandwiched” is in fact “sandwiched” � (𝑛𝑛𝑛𝑛� − 𝑛𝑛(𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟𝑟𝑟− 𝑛𝑛𝑟𝑟𝑟𝑟𝑟𝑟� (𝑛𝑛𝑛𝑛 −� 𝑛𝑛𝑟𝑟𝑟𝑟𝑟𝑟(𝑛𝑛𝑛𝑛� 𝑟𝑟𝑟𝑟𝑟𝑟 (𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟𝑟𝑟 � (𝑛𝑛𝑛𝑛𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟��� =𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟��� −=𝑟𝑟𝑟𝑟 −𝑟𝑟𝑟𝑟���� =𝑟𝑟𝑟𝑟,𝑟𝑟𝑟𝑟� −� ,𝑟𝑟𝑟𝑟=𝑟𝑟𝑟𝑟� −=𝑟𝑟𝑟𝑟� ,𝑟𝑟𝑟𝑟−� . � �=𝑟𝑟𝑟𝑟. −between� between. the inscribed thebetween inscribed and the circumscribed and inscribed circumscribed and circumscribed 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 𝑛𝑛𝑛𝑛 cylinders.𝑛𝑛𝑛𝑛 cylinders. As 𝑛𝑛𝑛𝑛 tends Ascylinders.𝑛𝑛𝑛𝑛 tends to in�inity, Asto in�inity,𝑛𝑛𝑛𝑛 tends the two the to stacksin�inity, two stacks of the twoof stacks of cylinderscylinders converge convergecylinders to the to form converge the formof the to of hemisphere. the the form hemisphere. of the hemisphere.

Figure 3.

FigureFigure 1. 1. Figure 1. FigureFigure 2. 2. Figure 2.

98 8 At RightAt Angles Right8 Angles | Vol.At Right|4, Vol. No. Angles 4,3, No.November |3, Vol. November 4, 2015 No. 3,2015 November 2015 Vol. 4,Vol. No. 4,3, No.November 3,Vol. November 4, 2015 No. 3,|2015 At November Right | At AnglesRight 2015 Angles | At Right98 Angles8 8 2 2At RightAt Right Angles2 AnglesAt∣ Vol. Right∣ 4,Vol. No. Angles 4, 3, No. November∣ 3,Vol. November 4, No. 2015 3, November2015 2015 Vol. 4, No. 3, November 2015 ∣ At Right Angles 3 The Method of Equilibrium

reveal how closely interlinked the disciplines Now I will discuss the second technique by which really are. Archimedes arrived at the same result for the To �ind the volume of a sphere by the Method of volume of a sphere. This technique, known as the Equilibrium, it helps to think of the solid as cut up Method of Equilibrium, was found in the lost into a large number of very thin strips hung end palimpsest. It sheds light on a very uniquely to end on an imaginary lever. This proof compares Archimedean way of thinking about surface areas the moments of two solids when placed on the and volumes of solid shapes, and employs an lever. Since volume is proportional to mass, argument that resonates with the modern notion moment of the solid can be de�ined as the product of integral calculus. of its volume and lever length (the distance from There has been considerable debate among the point about which the shapes are hung to the mathematicians about which Method of centroid of the volume). 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 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴. well before calculus emerged as a powerful Consider the cylinder and cone of revolution mathematical tool, so both methods contain ideas obtained by rotating rectangle 𝐴𝐴𝐴𝐴𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝐴𝐴𝐴𝐴 and triangle much ahead of their time. Historians of 𝐴𝐴𝐴𝐴𝑂𝑂𝑂𝑂𝐴𝐴𝐴𝐴 about the 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 axis. Suppose thin vertical mathematics such as Howard Eves argue that the slices of thickness Δ𝑥𝑥𝑥𝑥 are cut from the three solids Method of Exhaustion is “sterile” because its at distance 𝑥𝑥𝑥𝑥 from 𝐴𝐴𝐴𝐴. The approximate volumes of elegance is apparent only if the result is already theSphere: sections of each solid are deduced to be: known. While this is debatable, the Method of Equilibrium is unique for Archimedes’ use of The equation of the circular mechanics to prove a purely mathematical result. cross-section� � of the� sphere� is Archimedes himself is said to have preferred the (𝑥𝑥𝑥𝑥 𝑥𝑥 𝐴𝐴𝐴𝐴𝑥𝑥 +𝑦𝑦𝑦𝑦 𝐴𝐴𝐴𝐴𝐴𝐴 , i.e., 𝑦𝑦𝑦𝑦 𝐴𝐴 𝑥𝑥𝑥𝑥(𝐴𝐴𝐴𝐴𝐴𝐴 𝑥𝑥. 𝑥𝑥𝑥𝑥𝑥𝑥 Method of Exhaustion because he felt that it was Therefore the volume of revolution of the mathematically more rigorous. Perhaps this was slice of� sphere with thickness Δ𝑥𝑥𝑥𝑥 and height Cone: born out of his innate preference for pure 𝑦𝑦𝑦𝑦 is 𝜋𝜋𝜋𝜋𝑦𝑦𝑦𝑦 Δ𝑥𝑥𝑥𝑥 = 𝜋𝜋𝜋𝜋𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥� 𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥�𝑥𝑥𝑥𝑥. mathematics to mechanical inventions. However, The volume of revolution of the slice of� in the words of E.T. Bell, “To a modern all is fair in Cylinder:cone with thickness Δ𝑥𝑥𝑥𝑥 and height 𝑥𝑥𝑥𝑥 is 𝜋𝜋𝜋𝜋𝑥𝑥𝑥𝑥 Δ𝑥𝑥𝑥𝑥. love, war, and mathematics.” Maybe the equilibrium argument is considered more elegant The volume of revolution of the slice of today because there is something enchanting cylinder� with thickness Δ𝑥𝑥𝑥𝑥 and height 𝐴𝐴𝐴𝐴𝐴𝐴 is when borders between related disciplines melt to 𝜋𝜋𝜋𝜋𝐴𝐴𝐴𝐴 Δ𝑥𝑥𝑥𝑥.

Figure 3.

9 At Right Angles | Vol. 4, No. 3, November 2015 Vol. 4, No. 3, November 2015 | At Right Angles 9 Vol. 4, No. 3, November 2015 ∣ At Right Angles 3 If the slices from the sphere and the cone are Therefore, the moment about 𝑂𝑂𝑂𝑂 of the slices imagined to be stacked at 𝐴𝐴𝐴𝐴, they form a single of the cone and sphere is 4 times the moment point mass. Their combined moment about the about 𝑂𝑂𝑂𝑂 of the slice of the cylinder. When a point 𝑂𝑂𝑂𝑂 is given by: large number of such slices are added together, the following expression is Sum of volumes of slices of sphere and cone obtained: And other � × length 𝑂𝑂𝑂𝑂𝐴𝐴𝐴𝐴 𝑂𝑂 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 � 𝑂𝑂𝑂𝑂𝑂𝑂 � volume of sphere + volume of cone 𝑂𝑂 4𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑥𝑥 𝑂𝑂 4𝑂𝑂𝑂𝑂�volume of cylinder𝑥𝑥 memorable The moment about 𝑂𝑂𝑂𝑂 of the slice cut from the � Here, 4𝑂𝑂𝑂𝑂 is the length of the lever arm.�𝑂𝑂�푂𝑂𝑂 Since the cylinder (when its position is unchanged) is � volume of the cone is known� to be � and that given by: Part II of the cylinder to be 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑂𝑂 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 , we get: triples – feature � Volume of slice of cylinder 8𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 � � 𝑂𝑂𝑂𝑂𝑂𝑂 � volume of sphere + 𝑂𝑂 8𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑥𝑥 × distance from 𝑂𝑂𝑂𝑂 to slice 𝑂𝑂 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑥𝑥 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 3 � � ��� 𝑂𝑂 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑥𝑥 � References Therefore, the volume of the sphere is 𝑥𝑥 Men of Mathematics Introduction to the History of Mathematics 1. Bell, E.T. Math! Encounters. Dover with High Publications School Students (1946). 2. Eves, Howard. . Holt, Rinehart & Winston (1969). 3. Lang, Serge. . Springer-Verlag, 1985. Shailesh Shirali

n Part I of this article we had showcased the triple (3, 4, 5) by highlighting some of its properties and some con�igurations where it occurred naturally. We now attempt to extend this to other triples of consecutive integers. To begin with, we study AMRUTHA MANJUNATH did her A-levels at Centre for Learning, Bangalore, and is currently an undergraduate student at Ashoka University. She enjoys mathematics, dance, baking and photography. She can be contacted the two ‘siblings’ of (3, 4, 5), namely, the triples (2, 3, 4) and at [email protected]. (4, 5, 6). We start �irst with the elder sibling, (4, 5, 6). (We do need to show the older ones some respect, don’t we?) IThe triple 4, 5, 6 In Figure 1 we see a sketch of a triangle 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴GeoGebrawith sides 4, 5, 6 (with 𝑎𝑎𝑎𝑎𝑎𝑎6, 𝑏𝑏𝑏𝑏 𝑎𝑎�, 𝑐𝑐𝑐𝑐 𝑎𝑎�). Is there anything special about the triangle? Let’s do someB exploration using .

6 4

A 5 C Figure 1.

Keywords: Triangle, consecutive integers, triple, double angle, sine rule, cosine rule, Pythagoras

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