PYTHAGORAS of SAMOS
circa 580-500 B.C.
There are no know first hand accounts of Pythagoras life and work. His history is built on myth
and legend which makes it difficult to discern fact and fiction. What is certain is Pythagoras
studies the properties of numbers, the relationships between them and patterns they form with
his work being responsible for the first golden age of mathematics.[1 pp 6-7]
Pythagoras gained his mathematical skills traveling throughout the world, in particular from
Egyptian and Babylonian cultures. Both of these cultures went beyond using numbers as a
counting tool and could perform calculations to build sophisticated accounting systems, design
and engineer complicated structures, reconstruct boundaries that washed away in the annual
flooding of the Nile River. Pythagoras realized that Egyptians and Babylonians conducted
calculations by blindly applying formulas handed down from generation to generation that
would give the correct answer. Nobody had ever questioned them to explore the logic
underlying the equations.[1pp 6-7]
It is estimated that Pythagoras traveled for about twenty years of his life before returning to
Samos. In returning to the island he found a new political structure in place. The once liberal
island had fallen under the rule of the very conservative Polycrates, who asked Pythagoras to
join his court in an effort to silence him. Pythagoras lived in a cave on a remote part of the
island where he started a school with one pupil whose name was also thought to be
Pythagoras. He paid this person to be his student at first because the student was very
reluctant. As the student became more enthusiastic for knowledge he told the student he could
no longer afford to pay him, at that point the student offered to pay for his education. He
established the Semicircle of Pythagoras school for a short time but the political climate was
not correct in Samos to maintain it.[1 p 8]
Pythagoras went to the city of Croton in southern Italy where he found a very wealthy patron by
the name of Milo. Milo was more famous than Pythagoras gaining fame as an excellent athlete
who had been champion of the Olympic and Pythian Games a record twelve times. Milo sets
aside part of his house and provides Pythagoras with enough room to establish a school.
Pythagoras founds the Pythagorean Brotherhood who were over 300 aristocrats capable of
understanding his teachings. The curriculum consisted of four areas: arithmetica (number
theory), harmonia, geometria and astrologia.[1p 8]
The manner that Pythagoras organized his school was to divide students into two groups
acousti (listeners) and mathematici. The acousti would listen to his lectures for three years
behind a curtain then they could be initiated into the inner circle of the mathematici. The law
prevented women from attending public meetings, but at least 24 were admitted to his school.
He followed the custom of teachers of the east of passing on his lessons by word of mouth. He
did not seem to write down any of his teachings. Members of his community were bound not to
disclose to outsiders anything taught by Pythagoras or discovered by others in the
brotherhood.[2 pp 86-88]
At the age of 60 he married one of his students a women named Theno who was in her late
teens or early twenties. Reports conflict about her. Some say she was Milo's daughter, some
say Pythagoras and they were never married. It was certain she was a good student and
mathematician who continued to develop his work in her life.[2 pp 86-88]
Pythagoras and his followers believed that numbers were found throughout nature and the world
in developing an almost cult like following. What set them apart from others was the philosophy
that "knowledge is the greatest purification," and to them knowledge meant mathematics.[2 pp
86-88]
Music was a good example of how numbers were contained in nature. The Pythagoreans found
that all harmonious notes were found by making the strings on an instrument (such as a lyre) in
simple ratio with each other using whole numbers such as 2:1, 3:2 and 4:3.[2 pp 87-88]
Astronomy was also an area that was studied and certain numeric relationships were found.
Pythagoras held that the seven known planets (including sun and moon) where carried around
the earth by a crystal sphere. It was impossible for gigantic spheres to whirl endlessly without
generating noise the system created a celestial harmony, which Pythagoras alone among mortal
men could hear.
Pythagoras studied properties of numbers which would be familiar to mathematicians today,
such as even and odd numbers, triangular numbers, perfect numbers etc. However to
Pythagoras numbers had personalities which we hardly recognize as mathematics today:
Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or
ugly. This feeling modern mathematics has deliberately eliminated, but we still find overtones of
it in fiction and poetry. Ten was the very best number: it contained in itself the first four integers one, two, three, and four [1 + 2 + 3 + 4 = 10] - and these written in dot notation formed a perfect
triangle.
All ideas could be represented by numbers.
1 - Reason because reason could produce only one consistent body of truth
2 - Man
3 - Woman
4 - Justice because it could be produced by two equal numbers 2+2=4 and 2*2=4
5 - Marriage since it was the union of 2 and 3
6 - Creation
Even numbers beyond the first were considered feminine and earthly, odd numbers were
masculine and divine. [2 p 89]
Today we particularly remember Pythagoras for his famous geometry theorem. Although the
theorem, now known as Pythagoras's theorem, was known to the Babylonians 1000
years earlier he may have been the first to prove it. [3]
Here is a list of theorems attributed to Pythagoras, or rather more generally to the
Pythagoreans.
(i)
The sum of the angles of a triangle is equal to two right angles. Also the Pythagoreans
knew the generalization which states that a polygon with n sides has sum of interior
angles 2n-4 right angles and sum of exterior angles equal to four right angles.
(ii)
The theorem of Pythagoras - for a right angled triangle the square on the hypotenuse is
equal to the sum of the squares on the other two sides. We should note here that to
Pythagoras the square on the hypotenuse would certainly not be thought of as a number
multiplied by itself, but rather as a geometrical square
constructed on the side. To say that the sum of two squares is equal to a third square
meant that the two squares could be cut up and reassembled to form a square identical
to the third square.
(iii) Constructing figures of a given area and geometrical algebra. For example they solved
equations such as a (a - x) = x2 by geometrical means.
(iv) The discovery of irrationals. This is certainly attributed to the Pythagoreans but it does
seem unlikely to have been due to Pythagoras himself. This went against Pythagoras's
philosophy the all things are numbers, since by a number he meant the ratio of two whole
numbers. However, because of his belief that all things are numbers it would be a natural
task to try to prove that the hypotenuse of an isosceles right angled triangle had a length
corresponding to a number.
(v)
The five regular solids. It is thought that Pythagoras himself knew how to construct the
first three but it is unlikely that he would have known how to construct the other two.
(vi) In astronomy Pythagoras taught that the Earth was a sphere at the center of the
Universe. He also recognized that the orbit of the Moon was inclined to the equator of the
Earth and he was one of the first to realize that Venus as an evening star was the same
planet as Venus as a morning star. [3]
[1] Simon, Singh. Fermat's Enigma. Walker & Company New York 1997.
[2] Burton, David M. The History of Mathematics. McGraw Hill 2003.
[3] Web site: MacTutor History. http://turnbull.mcs.st-and.ac.uk/~history (9/2000).
One of the subjects the Pythagoreans studied were called figurative numbers. The
two types we will examine here are triangular and square numbers. The number of
symbols that can be represented by the numbers can arranged in a square as
shown below.
The first 4 triangular numbers t1=1, t2=3, t3=6 and t4=10



 

 
  
 
  
   
1
3
6
10
The first 4 square numbers s1=1, s2=4, s3=9 and s4=16
1
4
9
16
One of the key questions
that was asked was to
classify and describe all of
the triangular numbers.
The Pythagoreans realized
the following.
s1  t1
t1  1
t2  1  2
t3  1  2  3


tn  1  2  3    n
s2  t 2  t1
and
s3  t 3  t 2


sn  t n  t n 1
In the case of triangular numbers the Pythagoreans were able to classify them by using
an array model type of argument.



  
 
 
2=1×2
6=2×3
12=3×4
In general two triangular numbers
form a rectangle.
2tn  nn  1
  
   
  
  
20=4×5
nn  1
tn 
2
To determine the sum of the square numbers 12+22+32+...+n2 was difficult for the
methods of algebra known at the time. It was posed by them but is best known as
being resolved in the 1st century. It was realized that the sums of higher powers could
be constructed from the sums of lower powers. In the case of square numbers the
following rectangles were constructed.


 
12  22  1  1  2  2  11  2


12  1  (1  1) 1
1
2
    
    
    
    

 22  32  1  1  2  1  2  3  3  11  2  3
In general we can account for all the stars in the rectangle by adding up the squares
and what is left in each row and equating that with the dimensions of the rectangle.
We then apply the formula for
1
2

 22    n 2  1  1  2  1  2  3    1  2  3    n  n  11  2  3    n
1
2

 2 2  32    n 2  1121  2 221  3321    n n21  n  1 n n21
1  2  3    n   1  1  2  2  3  3    n  n  nn  1
1  2  3   n  1  2  3   n  1 2  3   n  nn 1
1  2  3    n     nn  1
1  2  3    n   nn  1n  1  
1  2  3    n   nn  12n  1
2
2
2
2
2
2
2
2
2
1
2
2
2
1
2
2
2
2
2
3
2
2
2
2
2
1 n n 1
2
2
3
2
2
2
2
2
1
2
3
2
2
2
2
2
1
4
2
2
1
2
1
2
1
2
2
1
2
12  2 2  32    n 2  16 nn  12n  1
1
2
2
The Pythagoreans also noticed some other interesting relationships built on the
geometric arrangement of numbers. An example of this would be the sum of odd
numbers are square numbers.

 
   
   
   
11
1 3  4
1 3  5  9
1  3  5  7  16

1  3  5  7  9  25  52
1  3  5    2n  1  n 2
In the further development of mathematics the problem of finding a formula for the
sums of powers of integers would become more important. This is especially true
when it came to develop the ideas associated with integral calculus.
In general mathematicians wanted to be able to find the sum of the kth powers of
the first n integers. In other words a formula for 1k+2k+3k+…+nk. A more modern
algebraic method was developed using the binomial theorem to expand sums. The
idea is that if you know all of the formulas for sum powers less than k you can use
those to develop a formula for the sum of the kth powers.
n
Find
j
j 1
k
Knowing
j j j
2
1
j 1
j 1
n
n
n
n
j 1
3
…
k 1
j

j 1
Here is how the sum of cubes (3rd powers) is developed knowing the following:
n
n
n
2
j
  16 nn  12n  1
 j  12 nn  1
1  n
1
j 1
j 1
j 1
To get the 3rd powers form a telescoping sum with the 4th powers.

 
 


n  1  0  2  1  3  2    n  n  1
4
4
4
4
n
4
4
4
4
  j 4   j  1
4
j 1

n

  j4  j4  4 j3  6 j2  4 j 1
j 1
4

Do some algebra to simplify the
expression you get. Notice that
the sum of the cubes are the
squares of the triangular
numbers.
j 1
  4 j3  6 j 2  4 j 1
n
4 j 3  n n 3  2 n 2  n
j 1
j 1
j 1
j 1
j 1

4 j 3  n 2 n 2  2 n  1
j 1
n
4 j 3  n 2 n  1
2
j 1
n
4 j 3  n 4  nn  12n  1  2nn  1  n

n
n
 4 j 3  nn  12n  1  2nn  1  n

j 1
n
 4  j 3  6  j 2  4 j   1
j 1

n
j 1
n

4 j 3  n n 3  2n 2  3n  1  2n  2  1
n
n

n
n
j
j 1
3
 14 n 2 n  1
2
The most famous result in Geometry is the Pythagorean Theorem. Much of modern mathematics developed
from this idea. This states in a right triangles the sum of the squares of the legs is equal to the square of the
hypotenuse. As mentioned before this was known before the time of Pythagoras, but he was tried to establish
this result in terms of proof. The algebraic methods to establish such a result were still primitive. Since the
Pythagoreans did not write down their knowledge the method could not be certain. People think it would
proceed using a dissection method as follows.
The areas of both of the squares can both be represented as the sum of the
areas of smaller squares and rectangles. The square on the left is 4 squares
and two rectangles so the area would be:
a2+b2+2ab
The square on the right is one square and a rectangle and the area would be:
c2+ 4 (ab/2)
If we equate these expressions we get:
a2+b2+2ab = c2+ 4 (ab/2)
The idea of "canceling" as we do in modern algebra is quite strange. What was
done was to deduct the area of the four triangles from both figures giving the
result:
a2+b2 = c2
Such proofs of addition of areas are so simple they may have been done by
other cultures. The Pythagoreans were interested in the study of arithmetic
relations. The problem of finding all right triangles whose sides were of integral
length is known as the Pythagorean problem.
The Pythagorean Theorem is an idea that relates the sides of a right triangle that
is thousands of years old. There are some words to identify certain sides of a right
triangle that are used. The hypotenuse of a right triangle is the side opposite the
right angle. The legs of a right triangle are the sides that form the right angle.
The Pythagorean Theorem
The Pythagorean Theorem states that the squares of the
legs of any right triangle added together is equal to the
square of the hypotenuse. Labeling the lengths of the legs as
a and b and the length of the hypotenuse as c we get the
relation: a2 + b2 = c2.
c
b
a
The justification for this (some people refer to this as a proof) has a long history.
Many different people have contributed to this and seeking logical justification of
the Pythagorean Theorem was one of the driving forces in the development of
mathematics. There are books filled with hundreds of arguments (proofs) as to why
the Pythagorean Theorem is true.
We will give one here. This is very old some people attribute it to Pythagoras
himself. The method used here is to compute the area in two different ways and
apply some algebra.
b
90
c
b
a
a
a
2
c
1
c
b
a
b
1. Start with any right triangle with sides
of length a, b (legs) and c (hypotenuse).
a
2. Make 4 identical (congruent) copies of
this triangle and place them vertex to
vertex to make a square.
1
3
c
c
b
4. Area of Big Square = (a+b)2
= (a+b)(a+b)
= a2 + 2ab + b2
5. Area of Big Square =
Area of 4 triangles
+ Area of small square
= 4·(½)·a·b
+ c2
= 2ab + c2
Set equal: a2+2ab+b2 = 2ab + c2
a2 + b2 = c2
3. Each angle of the inside figure is a
right angle since:
m1 + m2 + 90 = m1 + m2 + m3
90 = m3
4. Compute the area of the big square by
squaring the length of the side.
5. Compute the area of the big square by
adding the area of 4 triangles and the
little square.
6. The two expression both compute the
same area. Set them equal and cancel
the term 2ab from each side.
Pythagorean Problem: Find all positive integer solutions of the Pythagorean equation x2+y2=z2.
The ordered triple (x,y,z) satisfying the equation is called a Pythagorean triple.
Pythagoras expressed a partial solution of the problem as follows:
y=2n2+2n
x=2n+1
z=2n2+2n+1
It is not clear if he was the originator of this or not. The method he used to establish this was far
ahead of its time. He reasoned you could produce a square from the next smaller number by
using an algebraic fact.
(2k-1)+(k-1)2=k2
Consider the number 2k-1 as a perfect square. The number 2k-1 is odd and there are infinitely
many perfect squares that are odd. So this is possible at least. Let m2=2k-1 and solve for k.
k=(m2+1)/2
and
k-1=(m2-1)/2
This was not trivial do to the algebra methods of the time. If we substitute we get:
m2+((m2-1)/2)2=((m2+1)/2)2
This has solution:
Remember that x,y and z will only be integers if m is odd. Let m=2n+1.
x=m
y=(m2-1)/2
z=(m2+1)/2
x=2n+1
y=((2n+1)2-1)/2
z=((2n+1)2+1)/2
=(4n2+4n+1-1)/2 =2n2+2n
=(4n2+4n+1+1)/2 =2n2+2n+1