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Thales Kelly Dwyer Feb 5, 2014 Thales was born around 624 BC, and is believed to have died around 546 BC (Thales, 2014). Thales was born in Miletus, an ancient Greek Ionian city, to Examyes and Cleobuline, who belonged to the royal Phoenicia family (Thales, 2014). Although Thales was supposedly an engineer by trade, he was also considered a businessman, politician, philosopher, teacher, and mathematician. In addition to these, Thales is regarded as the first Greek philosopher by Aristotle, the father of science, and “the first true mathematician (Thales, 2014)”. With all of these titles, it is no wonder that Thales was recognized as the first of the Seven Sages (Thales, 2014; Biography of Thales, 2009). Thales lived a modest life, as it seems many philosophers did. This lifestyle was looked down upon by others. Thales set out to prove that he could use his intelligence to make money if he wanted to (Thales, 2014; O’Grady). One story of Thales’ business ventures involves the purchase and sale of the olive presses in Miletus. Thales is said to have predicted a particularly bountiful olive harvest one year. In anticipation of this event, Thales bid all of the olive presses in Miletus, and was able to get them at a low price. When the time came, the olive harvest was in fact plentiful. Thales was than able to rent out the presses for a considerable profit(Thales, 2014; Biography of Thales, 2009; Khalaf, 2014; O’Grady). Miletus had an expansive maritime trade system (O’Grady). Thales used the information he gathered in his studies of astrology to improve the trade system. He recognized that the constellation Ursa Minor was more practical for navigation than Ursa Major, which at the time was widely used (Khalaf, 2014; Biography of Thales, 2009; O’Grady). This would have been very valuable for the Greeks of Miletus. Thales also had a hand in politics. At the time, the Ionians were helping defend Anatolia from the Persians. Thales is said to have consulted on the matter. He urged the Ionian cities to form a union with a capital at Teos, a city in the center of Ionia (Thales, 2014; Khalaf, 2014). There is also a story in which the army was said to have come to a river, Halys, which they were not able to cross. Thales engineered a structure upstream that caused the water to change course, making it possible for the army to cross (Thales, 2014; Lendering, 2005). However, McKirahan (2011) states that these historical events would have occurred after Thales’ time. In a way, Thales was involved in the peace agreement between the Medes and Lydians, who had been at war for over five years. Thales predicted that a solar eclipse would occur in the year 585 BC (Thales, 2014; Biography of Thales, 2009; Lendering, 2005; Khalaf, 2014; O’Grady; McKirahan, 2011). In the fifth year of fighting, 585 BC, there was in fact a solar eclipse. It was recounted that on observation of the event, the two armies ceased fighting and came to a peace agreement (Khalaf, 2014; O’Grady). Predicting this event within the year was an impressive feat for the Greeks, not even Babylonian astronomers were able to predict with greater accuracy (Lendering, 2005) and each solar eclipse is visible only in a certain region making the pattern more difficult to establish than that of lunar eclipses (Khalaf, 2014). As a philosopher, Thales attempted to define the base of all life without the use of the Greek Gods (Thales, 2014; Biography of Thales, 2009; Khalaf, 2014). This was a revolutionary idea for his time; what’s more, Thales was the first philosopher to think this way (Lendering, 2005; Khalaf, 2014), making him “the founder of the school of natural philosophy” (O’Grady). Thales believed that the first principle was water, meaning all things came from water (Thales, 2014). According to Aristotle, Thales may have concluded this by observing that most living things have moisture (O’Grady). Thales believed that the Earth floated on water. He explained earthquakes in a way that supported this notion, saying that because Earth floats on water, and water moves, there are earthquakes (Khalaf, 2014). Although the theory that all things come from water has been disproved, the idea of taking mythology out of philosophy revolutionized Thales Kelly Dwyer Feb 5, 2014 science (Thales, 2014). As a highly esteemed individual and innovative philosopher, Thales was the considered the founder of the Ionian school of philosophy in Miletus (Biography of Thales, 2009). Thales taught Anaximander, the teacher of Pythagoras (Biography of Thales, 2009). As a teacher, Thales advised Pythagoras to visit Egypt to further his studies. Thales’ students were encouraged to pursue their own ideas, be critical of ideas, and open to discussion (O’Grady). This was is likely due to Thales’ methods of questioning and explaining, which laid the foundation for the scientific method (Biography of Thales, 2009; O’Grady). In addition to being known as a philosopher, Thales was known as a mathematician. He had a theoretical and practical understanding of geometry and is believed to have introduced the notion of logical proofs of abstract concepts(Biography of Thales, 2009). Eves (1990) states that “demonstrative geometry began with Thales” (p.72). That is, Thales was one of the first to ask why things were true, rather than how, which had previously been sufficient. At this time, deductive reasoning became and to this day remains a key component of mathematical thinking (Eves, 1990). In his studies, Thales traveled to Egypt, where he observed workers measuring property boundaries after the flooding of the Nile (O’Grady; Biography of Thales, 2009). While in Egypt, Thales is also said to have measured the height of a pyramid. To do this, Thales waited until the length of his shadow measured the same as his height, recognizing that the same proportion would hold true for the pyramid (Biography of Thales, 2009). He then introduced these ideas to Greece upon his return (Biography of Thales, 2009; Khalaf, 2014; O’Grady), calling it geometry meaning, “earth measure” (Biography of Thales, 2009). After returning to Greece, Thales converted what he had seen the land surveyors doing using stakes and ropes to create a system of points and lines (Biography of Thales, 2009). He continued to study geometry and is credited several important findings. First, Thales is credited with defining the diameter of a circle as its bisector. It is not exactly known how Thales proved this theory. McKirahan (2011) suggests that it may have been as simple as cutting out a circle and folding it over on itself. Thales also recognized that the base angles of an isosceles triangle and pairs of vertical angles had equal measures (O’Grady; Lendering, 2005; Biography of Thales, 2009). Again, it is not exactly certain how Thales’ proved that the base angles of an isosceles triangle were congruent. It is likely that Thales proved that vertical angles were congruent using a logical explanation involving straight angles (Figure 1). It was known that all straight angles were equal. Therefore, Thales knew that a c c b. Thus, Figure 1 vertical angles a and b are equal (Eves, 1990). Thales is also credited with the discovery of the angle-side-angle and angle-angle-side triangle congruencies (O’Grady; Lendering, 2005; Biography of Thales, 2009). It is believedthat Thales used these concepts to calculate the distance of ships from the shore (O’Grady). Knowing the distance of ships at sea was important to the city of Miletus. The tradesmen needed to know when shipments of goods were coming in to the city. The government in Miletus also benefited by knowing if enemies were approaching the shore (Biography of Thales, 2009). Figure 2 demonstrates a possible method for finding the distance. In this case, Thales is standing at point A looking out at the boat. Thales first would Figure 2 have constructed a segment AG perpendicular to his line of sight from the shore to the boat and Thales Kelly Dwyer Feb 5, 2014 a line perpendicular to AG at G. Next, he would have constructed the midpoint, H, of AG . From the midpoint, he could construct the angle between HA and the boat. Next, he could construct the vertical angle of H . By extending the rays of the angle and finding its intersection with the perpendicular line at G, we can find point I. By Thales’ understood theorem of angle-side-angle, he knew that ABH GIH . Thus, AB GI , giving Thales an easily measured distance on the shore (Ross, 2010). Lastly, Thales stated that any angle inscribed in a semicircle is a right angle. This final idea is known as Thales’ Theorem (O’Grady; Lendering, 2005; Biography of Thales, 2009). Thales used his knowledge of the fact that the sum of two right angles made a triangle and his previous theorems. Because AD, AB, AC are all radii of circle A, he know that they were congruent. This also meant that ABD and ABC were isosceles, which to Thales meant that their base angles were congruent. From here, it is not exactly sure how Thales proceeded. A modern proof of this theorem states that 2 180 , 2 180, and 180. we arrive at the conclusion that 90. Usingsubstitution, However, Thales would not have used this method because he had no notion of degrees in angle measures. As with many historical figuresof this time, it is difficult to know if the discoveries attributed to Thales are actually his. It is important to be critical of his so-called findings and mathematical discoveries. Although some ancient historians mention their existence, there is no physical evidence of Thales’ writings. The information gathered on Thales comes from the accounts of Greek historians, which may have been embellished. Many modern historians are doubtful of Thales’ involvement in many of the stories told. It may be that Thales’ did all of these things or only a few of them. The ideas could have been original, he could have learned about them from others and improved upon them, or he could have simply repeated what he learned from others. In any case, Thales was a contributor most importantly to philosophy and mathematics. Thales’ was the first to look at the world from a more modern scientific way, rather than a mythological way. Although his philosophy the base of all things is water was wrong, he relied on explanations that did not involve mythology. Thales looked for logical explanations and reasoning to his philosophical and mathematical theories. Scientists are still studying what the base of all things is. The mathematics attributed to Thales is still used today and taught in schools around the world. Thales would not accept things for what they appeared to be. He wanted to know what they were true. By rejecting the norms of society of his time, Thales laid the foundation for modern science and mathematics. Thales Kelly Dwyer Feb 5, 2014 Works Cited Biography of Thales. (2009). Retrieved 2014, 28-Jan from Math Open Reference: http://www.mathopenref.com/thales.html Eves, H. (1990). An Introduction to the History of Mathematics (Sixth Edition). Pacific Grove, CA: Thomson Learning, Inc. Khalaf, S. G. (2014). Thales of Miletus. Retrieved 2014, 28-Jan from http://phoenicia.org/thales.html Lendering, J. (2005). Thales of Miletus. Retrieved 2014, 28-Jan from Livius.org: http://www.livius.org/th/thales/thales.html McKirahan, R. D. (2011). Philosophy Before Socrates : An Introduction With Texts and Commentary (2nd Edition). Indianapolis, IN: Hackett Publishing Co. O’Grady, P. (n.d.). Thales of Miletus. Retrieved 2014 28-Jan from Internet Encyclopedia of Philosophy: http://www.iep.utm.edu/thales/ Ross, W. T. (2010, Sept 4). Thales. Retrieved February 5, 2014, from Nature of Mathematics: http://natureofmathematics.wordpress.com/lecture-notes/thales/ Thales. (2014). Retrieved 2014, 28-Jan from The Famous People: http://www.thefamouspeople.com/profiles/thales-263.php