Pythagoras and the Pythagoreans next up previous
Next: About this document

Pythagoras and the Pythagoreans

Historically, Pythagoras means much more that the familiar theorem about right triangles. The philosophy of Pythagoras and his school has impacted the very fiber of mathematics and physics, even the western tradition of liberal education no matter what the discipline.


Pythagorean  philosophy was the prime source of inspiration for Plato and Aristotle; the influence of these philosophers is without question and is immeasurable.

Pythagoras and the Pythagoreans

Little is known of his life. Pythagoras (fl 580-500, BC) was born in Samos on the western coast of what is now Turkey. He was reportedly the son of a substantial citizen, Mnesarchos. There he lived for many years under the rule of the tyrant Polycrates, who had a tendency to switch alliances in times of conflict -- which were frequent.

He met Thales, likely as a young man, who recommended he travel to Egypt. It seems certain that he gained much of his knowledge from the Egyptians, as had Thales before him.

Probably because of continual conflicts and strife in Samos, Pythagoras settled in Croton, on the eastern coast of Italy, a place of relative peace and safety.

Even so, just as he arrived, Croton lost a war to neighboring city Locri, but soon thereafter defeated utterly the luxurious city of Sybaris.

This is where Pythagoras began his society.

The Pythagorean School

tex2html_wrap_inline492 The school of Pythagoras was every bit as much a religion as a school of mathematics. For example, here are some of the rules:

Vegetarianism was strictly practiced probably because Pythagoras  preached the transmigration of soulsgif.

The school of Pythagoras represents the mystic tradition in contrast with the scientific!!

Indeed, Pythagoras regarded himself as a mystic and even semi-divine. Said Pythagoras

"There are men, gods, and men like Pythagoras."

It is likely that Pythagoras  was a charasmatic.

Life in the Pythagorean society was more-or-less egalitarian.

The Pythagorean Philosophy

The basis of the Pythagorean philosophy is simply stated:

"There are three kinds of men and three sorts of people that attend the Olympic Games. The lowest class is made up of those who come to buy and sell, the next above them are those who compete. Best of all, however, are those who come simply to look on. The greatest purification of all is, therefore, disinterested science, and it is the man who devotes himself to that, the true philosopher, who has most effectually released himself from the 'wheel of birth.'"gif

The message of this passage is radically in conflict with modern values. We need only consider sports and politics.

tex2html_wrap_inline492 Is not reverence these days is bestowed only on the ``super-stars"?

tex2html_wrap_inline492 Are not there ubiquitous demands for accountability!!

The Pythagorean Philosophy

The gentlemangif, of this passage, has had a long run with this philosophy, because he was associated with the Greek genius, because the ``virtue of comtemplation" acquired theological endorsement, and because the ideal of disinterested truth dignified the academic life.

The Pythagorean Philosophy ála Bertrand Russell

From Bertrand Russellgif, we have

"It is to this gentleman that we owe pure mathematics. The comtemplative ideal -- since it led to pure mathematics -- was the source of a useful activity. This increased it's prestige and gave it a success in theology, in ethics, and in philosophy."

Mathematics, so honored, became the model for other sciences. Thought became superior to the senses; intuition became superior to observation.

The combination of mathematics and theology began with Pythagoras. It characterized the religious philosophy in Greece, in the Middle ages, and down through Kant. In Plato, Aquinas, Descartes, Spinoza and Kant there is a blending of religion and reason, of moral aspiration with logical admiration of what is timeless.

Platonism was essentially Pythagorean ism. The whole concept of an eternal world revealed to intellect but not to the senses can be attributed from the teachings of Pythagoras.

The Pythagorean School gained considerable influence in Croton and became politically active -- on the side of the aristocrasy. Probably because of this, after a time the citizens turned against him and his followers, burning his house. Forced out, he moved to Metapontum, also in Southern Italy. Here he died at the ages of eighty. His school lived on, alternating between decline and re-emergence, for several hundred years.

Tradition holds that Pythagoras left no written works, but that his ideas were carried on by his eager disciples.

What is known of the Pythagorean school is from a book written by the Pythagorean, Philolaus of Tarentum. From this book Plato learned the philosophy of Pythagoras.

The dictum of the Pythagorean school was All is number.

What this meant was that all things of the universe had a numerical attribute that uniquely described them. For example,

Pythagorean Mathematics

One point: generator of dimensions.
Two points: generator of a line of dimension one
Three points: generator of a triangle of dimension two
Four points: generator of a tetrahedron, of dimension three.

The sum of these is ten and represents all dimensions. Note the abstraction of concept. This is a distance from ``fingers and toes".

Pythagorean Mathematics

Classification of numbers. The distinction between even and odd numbers certainly dates to Pythagoras. From Philolaus, we learn that

"...number is of two special kinds, odd and even, with a third, even-odd, arising from a mixture of the two; and of each kind there are many forms."
And these, even and odd, correspond to the usual definitions. But even-odd means a product of an even and odd number. Note: orginally the number 2 was not considered even.

Prime or incomposite numbers and secondary or composite numbers are defined in Philolaus:

Proposition. There are an infinite number of primes.

Proof. (Euclid) Suppose that there exist only finitely many primes tex2html_wrap_inline502 . Let tex2html_wrap_inline504 . The integer N-1, being a product of primes, has a prime divisor tex2html_wrap_inline508 in common with N; so, tex2html_wrap_inline508 divides N- (N-1) =1, which is absurd!

The search for primes goes on. Eratsothenes (276 B.C. - 197 B.C.), who worked in Alexandria, devised a seive for determining primes. This seive is based on a simple concept:

Lay off all the numbers, then mark of all the multiples of 2, then 3, then 5, and so on. A prime is determined when a number is not marked out. So, 3 is uncovered after the multiples of two are marked out; 5 is uncovered after the multiples of two and three are marked out.

The Primal Challenge

The search for large primes goes on: Below is a list of the largest found to date. For a great deal of information on primes, including the numbers below, check out the The Primes Home Page A special method, the Lucas-Lehmer test has been derived to check primality.

 2^859433-1 (258,716 digits); Slowinski and Gage, 1994
 2^756839-1 (227832 digits); Slowinski and Gage, 1992
 391581*2^216193-1 (65087 digits); Noll and others, 1989
 2^216091-1 (65050 digits); Slowinski, 1985
 3*2^157169+1 (47314 digits); Jeffrey Young, 1995 
 9*2^149143+1 (44898 digits); Jeffrey Young, 1995
 9*2^147073+1 (44275 digits); Jeffrey Young, 1995
 9*2^145247+1 (43725 digits); Jeffrey Young, 1995
 2^132049-1 (39751 digits); Slowinski, 1983
 9*2^127003+1 (38233 digits); Jeffrey Young, 1995

Subdivisions of even numbersgif are reported by Nicomachus (a neo-Pythagorean, ~100 A.D.).

tex2html_wrap_inline492  even-even -- tex2html_wrap_inline518
tex2html_wrap_inline492  even-odd -- 2(2m+1)
tex2html_wrap_inline492  odd-even -- tex2html_wrap_inline526

Similar subdivisions of odd numbers are:

tex2html_wrap_inline492  prime and incomposite -- ordinary primes excluding 2,
tex2html_wrap_inline492  secondary and composite -- ordinary composite with prime factors only,
tex2html_wrap_inline492  relatively prime -- two composite numbers but prime and incomposite to another number, e.g. 9 and 25.

Actually the third category is wholy subsumed by the second.

Also ascribed to the Pythagorean s is the study of perfect and amicable and deficient numbers.

A number n is perfect if the sum of its divisors is itself: Examples: ( 6, 28, 496, 8128, ...) In Euclid, we find the proposition: If tex2html_wrap_inline534 is prime, then tex2html_wrap_inline536 is perfect. (Try , p= 2, 3, 5, 7 to get the numbers above.)

The pair of numbers a and b are called Amicable If the divisors of a sum to b and if the divisors of b sum to a. Example: 220 and 284.

In addition, the number a was classified as abundant or deficient according as their divisors summed greater or less than a, respectively.

Example: 12-divisors are: 6,4,3,2,1-- tex2html_wrap_inline554 . So, 12 is abundant.

Example: All primes are deficient.

Mersenne Primes

Just the fact of finding perfect numbers using the previous propositions has spawned a cottage industry of determining those numbers p for which tex2html_wrap_inline534 is prime. Such primes are called Mersenne (1588-1648) primes after the friar of the 17 tex2html_wrap_inline560 century So far 33 have been found, though it is unknown if there is another between the 32nd and 33rd. It's not known if there are an infinity.



Mersenne Primes

number    p                        year  by
 1-5   2,3,5,7,13         in or before the middle ages
 6-7   17,19              1588  Cataldi
 8     31                 1750  Euler
 9     61                 1883  Pervouchine
10     89                 1911  Powers
11     107                1914  Powers
12     127                1876  Lucas
13-14  521,607            1952  Robinson
15-17  1279,2203,2281     1952  Lehmer
18     3217               1957  Riesel
19-20  4253,4423          1961  Hurwitz & Selfridge
21-23  9689,9941,11213    1963  Gillies
24     19937              1971  Tuckerman
25     21701              1978  Noll & Nickel
26     23209              1979  Noll
27     44497              1979  Slowinski  & Nelson
28     86243              1982  Slowinski
29     110503             1988  Colquitt  & Welsh jr.
30     132049             1983  Slowinski
31     216091             1985  Slowinski
32?    756839             1992  Slowinski & Gage
33?    859433             1994  Slowinski and Gage,

Figurate Numbers. Numbers geometrically constructed had a particular importance to the Pythagorean s.

Triangular numbers. These numbers are 1, 3, 6, 10, ... . The general form is the familiar



Square numbers These numbers are clearly the squares of the integers 1, 4, 9, 16, and so on. Represented by a square of dots, they prove(?) the well known formula



The gnomon is basically an architect's template that marks off "similar" shapes.

Note the gnomon has been placed so that at each step, the next odd number of dots is placed.

Figurate Numbers of any kind can be calculated.

The Pentagonal and Hexagonal numbers are shown in the graphs below.

external Note that the sequences have sums given by




Similarly, polygonal numbers of all orders are dsignated; this process can be extended to three dimensional space, where there results the polyhedral numbers. Philolaus is reported to have said:

All things which can be known have number; for it is not possible that without number anything can be either conceived or known.

The Pythagorean Theorem

Whether Pythagoras  learned about the 3, 4, 5 right triangle while he studied in Egypt or not, he was certainly aware of it. This fact though could not but strengthen his conviction that all is number. It would also have led to his attempt to find other forms. How might he have done this?

One place to start would be with the square numbers, and arrange that three consecutive numbers be a Pythagorean  triple! Consider for any odd number m,


which is the same as




Put the gnomon around tex2html_wrap_inline578 . The next number is 2n+1, which we suppose to be a square.


which implies


and therefore


It follows that


This idea evolved over the years and took other forms.

Did Pythagoras  or the Pythagorean s actually prove the Pythagoras  Theorem? Probably not. Although a proof is simple to give, the Pythagorean s had only a limited theory of similarity. And perhaps the reason was that rigor had not yet advanced to that level, at least in the early and middle period. The late Pythagorean s ( 400 B.C) however probably did supply a proof of this most famous of theorems.

Theorem I-47. In right-angled triangles, the square upon the hypotenuse is equal to the sum of the squares upon the legs.

external external

This figure is modelled after the original figure used by Euclid to prove the result. It is known to the French as pon asinorum and to the Arabs as the Figure of the Bride.

More Pythagorean  Geometry

Contributionsgif by the Pythagorean s include

From Kepler we have the quote

``Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel."

A line AC divided into extreme and mean ratio is defined to mean that it is divided into two parts, AP and PC so that AP:AC=PC:AP, where AP is the longer part.


Let AP=x and AC= a. Then the golden section is


and this gives the quadratic equation


The solution is


The golden sectiongif is the positive root:


And this was all connected with the construction of a pentagon.

The Pythagorean Pentagram

First we need to construct the golden section. The geometric construction, the only kind acceptedgif, is illustrated below.

Assume the square ABCE has side length a. Bisecting DC at E construct the diagonal AE, and extend the segment ED to EF, so that EF=AE. Construct the square DFGH. The line AHD is divided into extrema and mean ratio.


Verification. Note,




The key to the compass and ruler construction of the pentagon is the construction of the isoceles triangle with angles tex2html_wrap_inline612 and tex2html_wrap_inline614 . We begin this construction from the line AC. Note the figure


Divide a line AC into the `section' with respect to both endpoints. So PC:AC=AP:PC; also AQ:AC=QC:AQ. Bisect the line AC and construct on the perpendicular at this midpoint the point B so that AP=PB=QB=QC.

Define tex2html_wrap_inline618 PAB and tex2html_wrap_inline620 QPB. Then tex2html_wrap_inline622 . This implies tex2html_wrap_inline624 , and hence tex2html_wrap_inline626 . Solving for tex2html_wrap_inline628 we, get tex2html_wrap_inline630 . Since tex2html_wrap_inline632 PBQ is isoceles, the angle tex2html_wrap_inline634 QBP tex2html_wrap_inline636 . Now complete the line BE=AC and the line BD=AC and connect edges AE, ED and DC. Apply similarity of triangles to show that all edges have the same length. This completes the proof.

Regular Polygons

The only regular polygons known to the Greeks were the equilaterial triangle and the pentagon. It was not until about 1800 that C. F. Guass added to the list of constructable regular polyons by showing that there are three more, of 17, 257, and 65,537 sides respectively. Precisely, he showed that the constructable regular polygons must have


sides where the tex2html_wrap_inline640 are distinct Fermat primes. Recall, a Fermat prime is a prime having the form


Fermat ( 1630) conjectured that all numbers of this kind are prime.



Pierre Fermat (1601-1665), was a court attorney in Toulouse (France). He was an avid mathematician and even participated in the fashion of the day which was to reconstruct the masterpieces of Greek mathematics. He generally refused to publish, but communicated his results by letter.

Regular Polygons

Are there any other Fermat primes? Here's what's known to date.


By the theorem of Gauss, there are constructions of regular polygons of only 3, 5 ,15 , 257, and 65537 sides, plus multiples,


sides where the tex2html_wrap_inline640 are distinct Fermat primes.

The Pythagorean Theory of Proportion

Besides discovering the five regular solids, Pythagoras  also discovered the theory of proportion. Pythagoras  had probably learned in Babylon the three basic means, the arithmetic, the geometric, and the subcontrary (later to be called the harmonic).

Beginning with a>b>c and denoting b as the --mean of a and c, they are:


The Pythagorean Theory of Proportion

In fact, Pythagoras  or more probably the Pythagorean s added seven more proportions. Here are a few, given in modern notation:


Note: each of these is expressible in the notation of proportion, as above.



Allowing A , G, and H denote the arithmetic, geometric and harmonic means, the Pythagorean s called the proportion


the perfect proportion. The proportion


was called the musical proportion.

The Discovery of Incommensurables

This discovery is usually given to Hippasus of Metapontum ( tex2html_wrap_inline670 cent B.C.). One account gives that the Pythagorean s were at sea at the time and when Hippasus produced an element which denied virtually all of Pythagorean  doctrine, he was thrown overboard.

Theorem. tex2html_wrap_inline672 is incommensurable with 1.

Proof. Suppose that tex2html_wrap_inline674 , with no common factors. Then




Thusgif tex2html_wrap_inline692 , and hence tex2html_wrap_inline694 . So, a=2c and it follows that


whence by the same reasoning yields that tex2html_wrap_inline700 . This is a contradiction. width .1in height .1in depth 0pt

But is this the actual proof known to the Pythagorean s? Note: Unlike the Babylonnians or Egyptians, the Pythagorean s recognized that this class of numbers was wholly different from the rationals.

``Properly speaking, we may date the very beginnings of ``theoretical" mathematics to the first proof of irrationality, for in ``practical" (or applied) mathematics there can exist no irrational numbers."gif Here a problem arose that is analogous to the one whose solution initiated theoretical natural science: it was necessary to ascertain something that it was absolutely impossible to observe (in this case, the incommensurability of a square's diagonal with its side).

The discovery of incommensurability was attended by the introduction of indirect proof and, apparently in this connection, by the development of the definitional system of mathematics.gif In general, the proof of irrationality promoted a stricter approach to geometry, for it showed that the evident and the trustworthy do not necessarily coincide.

Other Pythagorean Geometry

Quadrature of certain lunes was performed by Hippocrates of Chios. He is also credited with the arrangement of theorems in an order so that one may be proved from a previous one (as we see in Euclid).

external The large lune ABD is similar to the smaller lune (with base on one leg of the right isosceles triangle tex2html_wrap_inline704 . Proposition:The area of the large lune ABD is the area of the semi-circle less the area of the triangle tex2html_wrap_inline704 .

next up previous
Next: About this document

Don Allen
Thu Feb 6 13:10:23 CST 1997