Pythagorean Theorem:
Euclid Style
By
Celia Robichaud and Leah Wofsy
Some history before Euclid
Plato
• Known for his enthusiasm about math and not as much his contributions to the
discipline
•Socrates was his mentor, Aristotle his student
•Founded the Academy in 387 B.C.
•“Let no man ignorant of geometry enter here”
Alexander the Great
•Established Alexandria in 332 B.C. in Egypt
Eudoxus (410355 B.C.)
• Pupil of Plato at the Academy
• Later became one of the greatest and most
influential mathematicians and astronomers.
• Provided complex explanations of lunar
and planetary movement using series of
concentric spheres
• First to provide mathematical explanation
for the movement of planets.
A Greek Mathematical View of the
Solar System... c.400B.C.
• The Sun and Moon had two spheres each, one rotated Westward
once every 24 hrs, one Eastward every month, one in the latitudinal
direction once per month.
•The five known planets each had five spheres explaining their motion.
• Eudoxus is remembered for two great theorems, the theory of
proportion and the method of exhaustion.
•Theory of Proportion: Eudoxus defines the meaning of equality
between two proportions, restoring the geometric theorems that had
before rested on the fact that any two magnitudes were
commensurable (now known to be untrue)
Theorem:
A/B = C/D iff for all integers m and n whenever mA < nB then
mC<nD, similarly for the relations > and =
• Found in Book V of the Elements
Method of Exhaustion
• Allowed mathematicians to determine the
areas of more complicated geometric figures,
leading to the determination of the area of a
circle (Ch.4)
•Approach an irregular figure by breaking it
down into pieces of known area
Euclid:
•Euclid came to Alexandria in 300 B.C. to set up school of Mathematics
•Wrote the Elements, which today, consists of 13 books
•The Elements was based off an axiomatic framework
Euclid’s Proof of
the Pythagorean
Theorem
• 300 B.C.
• Not an algebraic proof (A2=B2+C2), but
a geometric proof (square on the
hypotenuse is equal to the squares on the
sides containing the right angle.
•GA and AC on the same line
•Only time in proof that fact
<BAC is right is used
•Proved slender
triangles congruent
by SAS
•FB and AB equal
from construction
•BD = BC for same
reason
•<FBC=<ABD,
both=<ABC + 90°
• ΔABD and
rectangle BDLM
share same base
and fall within
same parallels
•Area of BDLM 2x
area ofΔABD
•Same goes for
ΔFBC and ABFG
Area BDLM=
2(ΔABD)=2(ΔFBC)=Area ABFG
Proved CELM=ACKH same way
by drawing AE and BK.
Therefore: Area of BCED
= BDLM + CELM
= ABFG + ACKH
Q.E.D.
But there’s more than one
way to prove a theorem…
•There are hundreds of proofs for the Pythagorean
Theorem, including one devised by James A. Garfield!
Knowing the area of a
trapezoid…
½(AB +AB + C2) = ½(B+A)(A+B)=½(A2 + B2 + 2AB)
Algebraic Proof
A angle and B angle are complementary
Blue central shape is a square with area C2
Area of whole shape is 4 (1/2 AB) + C2
Or
(A + B) 2 = A2 + 2AB + B2
A2 + 2AB + B2 = 4 (1/2 AB) + C2
C2 = A2 + B2
Converse proof
C
B
E
Prove that
D
A
BAC and DAC are congruent
CD2 = AD2 + AC2 = AB2 + AC2 = BC2
CD = BC
BAC and DAC are congruent by SSS
Non-Euclidean Geometry
“Alternative” Geometry discovered
by Gauss in early 19th C.
Maintains that triangles have less
than 180° in their angles
Reinforced by work of Johann
Bolyai and Nicolai Lobachevski
Bernhard Reimann, using
unbounded but finite lines, also
explained a geometric universe
where triangles can have more than
180°.
Questions to ponder •With so many possible ways to prove Pythagoras’ theorem, why did Euclid
choose to prove it with the method that he did?
•Why was the work in other disciplines at the time so incorrect but Euclid’s
work has stood the test of time?