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ARBELOS OF ARCHIMEDES EXPOSITORY PAPER Micki McConnell In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics Jim Lewis, Advisor July 2008 Arbelos of Archimedes 1 The term “arbelos” is a Greek word meaning “shoemaker’s knife”. Archimedes is credited with naming the geometric figure that he was first to study as the arbelos. It is stated that he played with this figure for fun and identified a number of interesting properties for the figure. In this paper, some information about Archimedes is given and a few of the arbelos’ properties and their proofs are examined. Archimedes was a Greek mathematician, physicist, engineer, inventor and astronomer. His greatest contribution to mathematics was in geometry with his work on the area of plane figures and volume and area of curved surfaces. He is considered to be one of the greatest mathematicians of antiquity and of all time. Archimedes was born around 287 BC in the seaport city of Syracuse, Sicily. At the time of his birth, Syracuse was a colony of Magna Graecia. His father was an astronomer named Phidias, and it was believed that Archimedes was related to King Hiero II, the ruler of Syracuse. There is no record to show if he was ever married or had children. Archimedes died around 211 or 212 BC when the city of Syracuse was captured during the Second Punic War. It is said that Archimedes was working on a mathematical diagram when a Roman soldier commanded him to come and meet General Marcellus. Archimedes refused to leave until he was done with the problem; this enraged the soldier, who then killed Archimedes with his sword. Archimedes’ last words were, “Do not disturb my circles.” He was referring to the circles in the mathematical drawing that he was working on at the time. It is believed that Archimedes studied in Alexandria, Egypt under Euclid. Some of his discoveries and inventions changed how people viewed and/or did things. A commonly cited anecdote regarding Archimedes explains how he invented a method for measuring the volume of an object with an irregular shape when the king asked him to find out if his new crown was solid Arbelos of Archimedes 2 gold. This discovery is known as Archimedes’ Principle and states that a body immersed in a fluid experiences a buoyant force equal to the weight of the displaced fluid. He also proved that the area of a circle was equal to multiplied by the square of the radius of the circle. Some of Archimedes’ writings include: The Measurement of a Circle, where he gave the value of 3 . The Quadrature of the Parabola, where he proved that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. The Sand Reckoner, where he set out to calculate the number of grains of sand that the universe could contain. The Sphere and the Cylinder, where he obtained the result of which he was most proud – namely, the relationship between a sphere and a circumscribed cylinder of the same height and diameter. Book of Lemmas or Liber Assumptorum, which was a treatise with fifteen propositions on the nature of circles. It is believed that this is actually based on an earlier work by Archimedes that is now lost. One idea that caught the attention of Archimedes was the arbelos. It is said that Archimedes was the first to study this figure. Even though the surviving works of Archimedes do not mention the arbelos, it is claimed that he studied and named the arbelos in his Book of Lemmas. An arbelos is a plane region bounded by a semicircle of diameter 1; connected to semicircles of diameters r and (1-r), all oriented the same way and sharing a common baseline. (The baseline of a semicircle is a straight line forming the diameter which connects the ends of an arc.) The figure has a number of interesting properties; many first identified by Archimedes. Archimedes’ idea entailed two smaller circles outside of each other but inside a third larger circle. Each circle was Arbelos of Archimedes 3 tangent to the others and their centers were along the same straight line. He wanted to find the area that was inside the larger circle, but outside the two smaller circles. Archimedes named the arbelos using one-half of the circle, either the top half or the bottom half. It is important to note that the arbelos is not represented by the circle being cut in half vertically, only horizontally (see Figure 1). Figure 1: the arbelos The geometric properties of the arbelos have a wide range, one being that the length of the lower boundary of the arbelos equals the length of the upper boundary. The proof for this comes from the knowledge that the circumference of a circle is proportional to its diameter. A more formal statement and proof occurs in the work of Behnaz Rouhani (2002): Arc length along the enclosing semicircles is the same as the arc length along the two smaller semicircles, i.e. Arc length AKB = Arc length AEC + Arc length CFB. To prove this, let us look at the figure below. Arbelos of Archimedes 4 As noted in the previous figure, point M is the center of the semicircle AKB. Point N is the center of the semicircle AEC and point P is the center of semicircle CFB. From this figure, since AM = a, and AN = b then we can say NM = a – b. Next, using semicircle CFB, we let CP = d so we can conclude that MC = a – 2d. We know that the circumference of a semicircle is given by the formula C = r . Therefore, we can now make some equations dealing with arc length that we will number for use in the sequel. Arc length AKB = a (1) Arc length AEC = b (2) Arc length CFB = d (3) Also from the previous figure we know that: NM + MC = AN (a-b) + (a-2d) = b 2a – 2d = 2b OR 2a = 2d + 2b which simplifies to a = d + b Now, using the information from (2) and (3): AEC + CFB = b d (b d ) a AKB A more sophisticated property deals with the area of the arbelos. This property states that the area of the arbelos equals the area of the circle whose diameter, CD, is the portion inside the arbelos of the common tangent line to the two smaller semicircles at their point of tangency, C (see Figure 2). In his Book of Lemmas, Archimedes devoted three of his fifteen Propositions to the study of the arbelos. This property is Proposition 4. The other two are Propositions 5 and 6. Arbelos of Archimedes 5 . A C B Figure 2: An area property of the arbelos In one proof of this proposition we reflect across the line through points A and B and observe that twice the area of the arbelos is what remains when the areas of the two smaller circles with diameters AC and CB are subtracted from the area of the larger circle with diameter AB. Since the area of a circle is proportional to the square of the diameter, the problem reduces to show that 2(CD)2 = (AB) 2 – (AC) 2 – (CB) 2. Since the length of segment AB equals the sum of the lengths of segment AC and segment CB, the equation can be simplified algebraically to (CD) 2 = (AC)(CB). The claim that the length of the segment CD is the geometric mean of the lengths of the segments AC and CB can be made. Now, if we imagine the triangle ADB being inscribed in a semicircle, it would have a right angle at point D and that consequently segment CD is “a mean proportional” between segments AC and CB. This proof may be found given as a “proof without words” by Roger B. Nelson (see Appendix A). Here is another statement and proof that can be found in the work of Behnaz Rouhani (2002): The area of the circle with diameter DC is the same as the area of the arbelos. For this proof, we use the diagram below. Draw the perpendicular segment DC from the tangent of the two semicircles to the edge of the largest semicircle. We claim that the area of the circle with diameter DC is the same as the area of the arbelos. Arbelos of Archimedes 6 C 1-r B For this diagram, let AB = 1, and AC = r. Using the Pythagorean Theorem, we can write the following equations that are numbered for future use. In triangle ADC: r2 + h2 = x2 (4) In triangle BDC: (1-r) 2 + h2 = y2 (5) In triangle ADB: x2 + y2 = 1 (6) When solving equations (4) and (5) simultaneously, and after much simplification, we arrive at the following equation. (1-r) 2 + x2 – r2 = y2 OR 1 – 2r + x2 = y2 (7) Now substituting equation (7) into equation (6), we get: x2 + 1 – 2r + x2 = 1 2x2 = 2r x= r Next, we substitute the above expression for x into both equations (7) and (4), and find corresponding expressions for y and h. y = 1 r and h= r r2 Arbelos of Archimedes 7 We know that the radius of the circle with center at point O is 1 r r 2 . Therefore, the area of 2 the circle with center at point O is given by equation (8). 1 ( r r 2 )2 4 r r2 A 4 4 A (8) From the above figure, we can tell that the area of the arbelos is the area of the semicircle with diameter AB minus the sum of the areas of the semicircles with diameters AC and BC. Thus, we can write, the following equation. 1 r 2 1 (1 r ) 2 A 8 2 2 2 2 A (1 r ) 2 8 8 2 r ( 2 r r 2 ) A 8 2 2 r 2 r A 8 r r2 r r2 A OR A 4 4 4 8 r2 Therefore, the area of the arbelos is the same as the area of the circle with diameter DC as shown in equation (8). With modern mathematical knowledge and notation, we are able to give proofs that would not have been available in Archimedes’ time. For our first identity, we know that all circles are similar and that the circumference of a circle is 2 times the radius, so the circumference of a semicircle would be times the radius. Also, we are able to establish the relationship using a circle of radius one. Arbelos of Archimedes 8 1 x [ 1+x 1-x ] The circumference for the larger semicircle would be times 1 or . The identity states that the two smaller semicircle lengths will equal the arclength of the larger semicircle. Thus, we need: = ( 1 2 x ) + ( 1 2 x ). Factor out , we want 1 = 1 x 1 x + 2 2 The x’s cancel out and so our desired identity is equivalent to, 1 = ½ + ½ . The second identity is that the area of the circle with diameter DC (see page 6) is the same as the area of the arbelos. We will use the following figure for our proof. . Arbelos of Archimedes 9 (x,y) .C A1 A2 (x,0) Again, we will assume that the larger circle has radius 1. Thus, we know that x2 + y2 = 1 hence y = 1 x 2 or y2 = (1-x2) . Also, since the area of a circle of radius r is r2, the area of a semicircle is ( r2 )/2. In our picture, A1 and A2 are semicircles and C is a circle with center at (x, y/2) and radius y/2. In order to show that the area of C equals the area of the Arbelos, we need to establish that 1/2 = A(A1 ) + A(A2 ) + A(C) or 1 x 2 1/2 = 1/2 ( ) 2 ( ) 1 x 2 + 1/2 ( ) 2 Factoring out , we need to establish that ½ = ( )+ ( 1 ( x) 2 4 ) (1 x) 2 (1 x) 2 1 ( x) 2 + + . 8 8 4 1 2x x 2 1 2x x 2 1 x2 The right hand side equals + + . 4 8 8 If we get a common denominator, 2x will cancel out and we will get: 1 x 2 1 x 2 2 2x 2 4 1 as desired. 8 8 2 Even though Archimedes was believed to be the first to study the arbelos, others after him investigated its properties as well. Pappus of Alexandria discusses the arbelos in Book IV of Arbelos of Archimedes 10 the Collection. Pappus does not cite Archimedes, but he also does not hold any claim to the most famous theorem about the arbelos. The theorem states that a chain of inscribed circles, Cn, in an arbelos has the following property. The distance from the diameter of the largest semicircle to the center of the nth circle in the chain, Cn, is exactly equal to dn, where dn is the diameter of Cn. Another person associated with the arbelos is Otto Mohr, a German civil engineer and professor of mechanics. Since the arbelos is a classical example of what is known as pure mathematics, it is well known in textbooks on solid mechanics under the name “Mohr’s circles.” The arbelos was and is studied by mathematicians throughout the world. The arbelos satisfies a number of unexpected identities of which the author only chose to evaluate two. Properties of the arbelos can be examined and proven in many different ways, and it ought to be mentioned that Archimedes enjoyed playing with the arbelos for fun. Certainly, for one to completely understand all the known properties of the arbelos would take a vast amount of time. However, even the introduction given here seems both valuable and extremely interesting. Arbelos of Archimedes 11 REFERENCES Archimedes. (n.d.) Retrieved June 19, 2008, from http://library.thinkquest.org Archimedes. (n.d.) Retrieved June 19, 2008, from http://www.cs.drexel.edu Arbelos-The Shoemaker’s Knife. (n.d.) Retrieved June 19, 2008, from http://www.cut-theknot.org Proof Without Words: The Area of an Arbelos, Nelsen, R Mathematics Magazine, 2002, vol. 75, no. 2, pp. 144 “Reflections on the Arbelos” HP Boas, American Mathematical Monthly, 2006, vol. 113, no. 3, pp. 236-249 “The Arbelos” B Rouhani, (2002) Retrieved June 19, 2008, from http://jwilson.coe.uga.edu Weisstein, Eric W. “Arbelos.” From MathWorld-A Wolfram Web Resource. http://mathworld.wolfram.com/Arbelos.html Arbelos of Archimedes 12 Appendix A Proof Without Words: The Area of an Arbelos -ROGER B. NELSEN LEWIS & CLARK COLLEGE THEOREM. Let P, Q, and R be three pints on a line, with Q lying between P and R. Semicircles are drawn on the same side of the line with diameters PQ, QR, and PR. An arbelos is the figure bounded by these three semicircles. Draw the perpendicular to PR at Q, meeting the largest semicircle at 5. Then the area A of the arbelos equals the area C of the circle with diameter QS [Archimedes, Liber Assumptorum, Proposition 4]. A=C Q Proof. A A1 A2 B1 B2 Bi Ai Ci A A1 A2 A!Ci A2 C2 C2