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(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,
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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
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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-
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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, mACB = mMCB + mACM = 90. (Bergemann, 2009) Thales used the
following two facts in finding this theorem: the sum of the angles in a triangle is equal to
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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.
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Bibliography
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O’Grady, P. (2001, September 17). Thales of Miletus. In Internet Encyclopedia of
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Thales of Miletus. (2011). In Wikipedia. Retrieved February 10, 2011 from
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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