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1 (student name omitted) Mr. Buck Algebra 1 29 March 2011 The Legacy of Thales Thales of Miletus was born around the year 624 B.C. in Miletus, Asia Minor, currently Balat, Turkey, and lived until about 546 B.C. (Thales of Miletus, 2011) He was the son of Examyas and Cleobuline, who were distinguished Phoenicians. (Lahanas, 2002) Thales was the first known Greek philosopher, scientist, and mathematician, although he was an engineer by occupation. (Knierim, 1999; O’Connor & Roberton, 1999) He was the teacher of Anaximander, who then was the teacher of Pythagoras. Thales was the first natural philosopher in the Milesian School, a school founded in the sixth century B.C. introducing new opinions different to the current viewpoint on how the world was organized. (Thales of Miletus, 2011) Thales was considered one of the Seven Sages of Greece, or Wiseman of the early sixth century B.C. There is little known about Thales’ personal life, partly because none of Thales’ writing survives, which makes it difficult to determine his views or to be certain about his mathematical discoveries. (O’Grady, 2001) However, in many textbooks on the history of mathematics, Thales is credited with five propositions of elementary geometry, one of which is referred to as Thales’ Theorem: i. A circle is bisected by any diameter, ii. The base angles of an isosceles triangle are equal, iii. The angles between two intersecting straight lines are equal, 2 iv. Two triangles are congruent if they have two angles and one side equal, v. An angle in a semicircle is a right angle (O’Connor & Roberton, 1999) The first proposition of elementary geometry credited to Thales is, “A circle is bisected by any diameter”. The formal definition, which Proclus wrote but attributed to Thales, is as follows: “A diameter of the circle is a straight line drawn through the centre and terminated in both directions by the circumference of the circle; and such a straight line also bisects the circle.” (O’Grady, 2001) Thales is credited with defining the diameter of a circle. Let r be the radius and d be the diameter. Then d = 2r. This result is the formula for the diameter of a circle. When any diameter bisects a circle, the circle is split into two equal or congruent parts. In the diagram below, the diameter of the circle is labeled d. (Lahanas, 2002) The diameter is an important concept in geometry. It helps in finding the radius, area and circumference of any circle. The bisection of a circle is also important in normal life, such as when folding paper or slicing pizza. (Allen, 1999) The second proposition of elementary geometry credited to Thales is, “The base angles of an isosceles triangle are equal.” An isosceles triangle is a triangle with at least 3 two congruent sides. Two sides are congruent if they have the same shape and size. In the figure below, the two equal sides have length b and the remaining side has length a. (Weisstein, 2011) This property states that the marked angles are congruent. Therefore an isosceles triangle has both two equal sides and two equal angles. (Weisstein, 2011) Proclus also wrote this theorem and gave credit to Thales. (O’Grady, 2001) The third proposition of elementary geometry credited to Thales is, “The angles between two intersecting straight lines are equal.” Therefore, let γ and be a pair of vertically opposite angles, as a and are a pair of vertically opposite angles. (Angles, 2011) According to Thales’ proposition, a = and γ = . (Lahanas, 2002) With this proposition, vertical angles in two intersecting straight lines are congruent. The fourth theorem of elementary geometry credited to Thales is, “Two triangles are congruent if they have two angles and one side equal.” Thales applied this theorem to determine the height of a pyramid. At 481 feet, the Great Pyramid was the tallest man- 4 made structure in the world for over 4,000 years, the longest period of time ever held for such a record. (How, 1997) The Great Pyramid was already over two thousand years old when Thales visited Giza, but its height was still unknown. By measuring the pyramid’s shadow when the shadow was the same height as the pyramid, Thales found a way to determine the height of the pyramids. He introduced the concept of ratio, and recognized its application as a general principle. (O’Grady, 2001) Possibly the most interesting and known theorem accredited to Thales is referred to in a passage of Lives of Eminent Philosophers by Diogenes Laertius. (O’Grady, 2001) Thales’ Theorem states that the angle inscribed in a semi-circle is a right angle. Thales was so impressed with this discovery that he decided to sacrifice an ox immediately after the idea came to him to symbolize the “strength” of the observation. (Thales’ Theorem, 2011) It is based upon his previous discoveries about the diameter and the properties of the isosceles triangle. In the diagram below, let M be the center of the circle through A, B, and C. (Bergemann, 2009) Then AM = BM = CM, and thus the triangles AMC and BMC are isosceles triangles. Therefore, mACB = mMCB + mACM = 90. (Bergemann, 2009) Thales used the following two facts in finding this theorem: the sum of the angles in a triangle is equal to 5 two right angles (180°), and the base angles of an isosceles triangle are equal. Thales' Theorem is a special case of the Inscribed Angle Theorem. (Thales’ Theorem, 2011) Although this might seem a simple observation, Thales was the first one who stated it and thus started what is now generally known as “deductive science”, the process of deriving suppositions and mathematical statements from observation by means of logic. (Knierim, 1999) Thales most likely acquired the basics of geometry from his time in Egypt. However, the evidence is that the Egyptian skills were in orientation, measurement, and calculation. Thales’ unique ability was with the characteristics of lines, angles, and circles. He recognized, noticed, and understood certain principles, which he probably “proved” through repeated demonstration. (O’Grady, 2001) Thales was a very brilliant and important mathematician. Not only did he create five propositions of elementary geometry, he crafted the way mathematics is done and used today. Without the ingenuity of Thales in the 7th century B.C., the world would be a different place. That is the legacy Thales left behind. 6 Bibliography Allen, D. (1997, February 3). Thales of Miletus. In Texas A&M University. Retrieved February 11, 2011 from http://www.math.tamu.edu/~dallen/history/thales2/thales2.html Angles. (2011). In LabSpace. Retrieved February 10, 2011, from http://labspace.open.ac.uk/mod/resource/view.php?id=420259 Bergemann, C. (2009, June 9). Proof of Thales’ Theorem. In PlanetMath. Retrieved February 10, 2011, from http://planetmath.org/encyclopedia/ProofOfThalesTheorem.html How Tall? (1997). In NOVA. Retrieved March 28, 2011, from http://www.pbs.org/wgbh/nova/pyramid/geometry/height.html Knierim, T. (1999, October 1). Thales. Retrieved February 11, 2011, from http://www.thebigview.com/greeks/thales.html Lahanas, M. (2002). Thales. Retrieved February 10, 2011 from http://www.mlahanas.de/Greeks/Thales.htm O’Connor, J. & Roberton, E. (1999, January). Thales of Miletus. In University of St. Andrews, Scotland. Retrieved February 10, 2011 from http://www-history.mcs.st-andrews.ac.uk/Biographies/Thales.html O’Grady, P. (2001, September 17). Thales of Miletus. In Internet Encyclopedia of Philosophy. Retrieved February 10, 2011 http://www.iep.utm.edu/thales/#SH9a Thales of Miletus. (2011). In Wikipedia. Retrieved February 10, 2011 from http://en.wikipedia.org/wiki/Thales Thales’ Theorem. (2011). In Wikipedia. Retrieved March 28, 2011 from 7 http://en.wikipedia.org/wiki/Thales%27_theorem Weisstein, E. (2011). Isosceles triangle. In Wolfram MathWorld. Retrieved February 10, 2011 from http://mathworld.wolfram.com/IsoscelesTriangle.html