Download Pythagoras and His Theorem Historical Context: Suggested

Pythagoras and His Theorem
Historical Context:
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When: ca. 500 B.C.
Where: Samos, Greek Ionia
Who: Pythagoras
Mathematics focus: Investigation of a useful property of right triangles.
Suggested Readings:
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Pythagoras and his contributions to mathematics and philosophy:
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Pythagoras.html
• Early Greek geometry:
http://jwilson.coe.uga.edu/EMAT6680/Greene/EMAT6000/greek%20geom/greekg
eom.html
• NCTM’s Historical Topics for the Mathematics Classroom (1969):
“Demonstrative geometry” (pp. 170-172) and “Early Greek geometry” (pp. 172174).
• Key search words/phrases: Pythagoras, Greek geometry, right triangle,
Pythagoreans
Problem to Explore:
Investigate special properties of a right triangle.
Why This Problem is Important:
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One of most famous geometrical theorems shared with students.
Introduces many interesting extensions and connections that are relatively
unknown.
Problem Solving Experiences:
Pythagorean Theorem: In a right triangle, the square of the hypotenuse is equal to
the sum of the squares of the other two sides.
a
c
b
Though well-known and often recited as a memorized litany with little understanding.
Students unfortunately remember only the phrase “a2 + b2 = c2” and do not “see” any
of the geometry involved (e.g. a relationship between areas). Yet, the Pythagorean
Theorem encompasses many other interesting aspects and connections. Though
known and used prior to Pythagoras’ time, the Theorem is credited to Pythagoras by
what Heath (1956) calls “tradition,” perhaps because it is often connected to the
Pythagorean proof of the necessary existence of irrational numbers.
Though Pythagoras is incorrectly credited with the Theorem’s “first” proof, Proclus, a
Greek philosopher and commentator on Euclid’s Elements, claimed that Pythagoras
“sacrificed an ox in honor of his discovery.” No definite Pythagorean-based proof is
known, except for speculations. The “usual” proof is driven by this diagram:
H
I
J
G
B
x
K
c-x
a
A
F
C
D
b
E
1. Prove the Pythagorean Theorem using this diagram and properties of similar
triangles.
In The Elements, the diagram for Euclid’s proof of the Pythagorean Theorem is
similar, but is necessarily different. That is, Euclid’s proof occurs as the conclusion to
Book 1, prior to his discussion in Book VI on similar triangles and proportionality of
corresponding sides. Euclid’s “alternative” diagram is:
H
I
J
G
B
x
c-x
K
a
A
F
C
b
D
E
2. Prove the Pythagorean Theorem using this diagram and the property that the
ratio of the areas of a parallelogram and a triangle with the same base and
height is 2:1.
In the real world, the Converse of the Pythagorean Theorem is more useful than the
Theorem. With a history that begins with the “3-4-5 rope stretchers” in Egypt, the
converse is used regularly by carpenters to ensure right angles while framing.
3. Prove the Converse of the Pythagorean Theorem: In a triangle, if the square of
the longest side is equal to the sum of the squares of the other two sides, the
triangle has a right angle.
Extension and Reflection Questions:
Extension 1: Many different “Behold!” proofs of the Pythagorean Theorem exist,
where the reasoning is evoked by reasoning directly from a picture. For each
example based on a right triangle with side lengths a, b, and c, explain the reasoning
involved:
• This is the most common “Behold!” proof:
b
b
a
b
b
c
a
a
a
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A “Behold!” proof attributed to both the Indian mathematician Bhāskara (1114–
1185) and the Chinese text Chou Pei Suan Ching (ca. 500 BC-200 AD):
c
a
b
c
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This proof seems cluttered, but then the “Behold!” occurs:
a
c
a
c
b
b
Extension 2: James A. Garfield, the twentieth President of the United States,
supposedly discovered this proof in 1876 while yet an Ohio State Representative and
doodling during a congressional session. He supposedly shared it with his fellow
Senators in the Congresional Halls during a recess. The proof eventually was
published in the New England Journal of Education (1876, Vol. 3, p. 161):
c
b
c
a
b
a
Prove that a2 + b2 = c2. Hint: Think “trapezoid.”
Open-ended Exploration: Try to prove the Pythagorean Theorem using other
mathematics that you know, such as trigonometry (be careful!), analytic geometry,
linear algebra, etc. Is any specific proof better than any others? Is there a “most
unusual” proof?