The Pythagorean Theorem
in different Civilizations
Christine Larson & Ashley Alex
• What Civilizations?
– Greeks, Chinese, Indians
• How was it approached?
• What techniques were used to prove it?
• How did it effect the rest of Mathematics?
Some Greek Methods…
•Contributed in the respect of Geometry
and formal proof
•Euclid’s Elements
• abandoned algebra (slogan), focused more
on geometry
•Led to geometric algebra, Thales boat and
shore problem, heights of structures, ratios,
distances related to astronomy such as
measuring the distance to the sun, distance
between the moon and sun and used to
solve equations with two unknowns
Some Indian Methods…
• Observed in the Sulba
Sutras; also used a
cutting and rotating
method (very similar to
Chinese proof)
• Used as an explanation
of how to construct
squares from original
figures, find heights by
use of shadows
• Greater shift towards
solving systems, linear
congruencies,
trigonometry
http://www.mathsisfun.com/pythagoras.html
Some Chinese Methods…
• Made a convincing
argument without using
an axioms (both Zhao
Shuang and Liu Hui)
• Focused more on
estimation, solving
equations, and Chinese
Remainder Theorem
• Helped solve problems
with similar triangles,
measure certain depths or
other measurements (like
homework problem)
Claim: c2 = (a-b)2 + 2ab
Cut and Rotate method…
Chinese
Indian
algebraic
Convincing
arguments;
“theorein:” to look
at
cut and rotate
concrete examples
“empirical” proof
a2 + b2 = c2
Astronomical
applications
Chapter 9
similar to
Diophantus’s
Arithmetica
Shared a
geometric
focus
Developed
constructions for
“completing the
square”
axiomatic proof system;
organized
Deductive reasoning
no numbers
Greeks
• Collection of 84 proofs…
http://www.cut-the-knot.org/pythagoras/index.shtml
• Wizard of Oz
http://www.youtube.com/watch?v=DUCZXn9RZ9s
• Chinese
http://www.davis-inc.com/pythagor/proof2.html