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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
The school of Pythagoras was every bit as much a religion as a school of mathematics. For example, here are some of the rules:
- To abstain from beans.
- Not to pick up what has fallen.
- Not to touch a white cock.
- Not to stir the fire with iron.
- Do not look in a mirror beside a light.
Vegetarianism was strictly practiced probably because Pythagoras preached the transmigration of souls.
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 school regarded men and women equally.
- They enjoyed a common way of life.
- Property was communal.
- Even mathematical discoveries were communal and by association attributed to Pythagoras himself -- even from the grave. Hence, exactly what Pythagoras discovered personally is difficult to ascertain.
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.'"
The message of this passage is radically in conflict with modern values. We need only consider sports and politics.
Is not reverence these days is bestowed only on the ``super-stars"?
Are not there ubiquitous demands for accountability!!
The Pythagorean Philosophy
The gentleman, 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 Russell, 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,
- The number one : the number of reason.
- The number two: the first even or female number, the number of opinion.
- The number three: the first true male number, the number of harmony.
- The number four: the number of justice or retribution.
- The number five: marriage.
- The number six: creation
- The number ten: the tetractys, the number of the universe.
- 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.
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:
- A prime number is rectilinear, meaning that it can only be set out in one dimension. The number 2 was not originally regarded as a prime number, or even as a number at all.
- A composite number is that which is measured by some number. (Euclid)
- Two numbers are prime to one another or composite to one another if their greatest common divisor is one or greater than one, respectively.
Proposition. There are an infinite number of primes.
Proof. (Euclid) Suppose that there exist only finitely many primes . Let . The integer N-1, being a product of primes, has a prime divisor in common with N; so, 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 numbers are reported by Nicomachus (a neo-Pythagorean, ~100 A.D.).
- even-even --
- even-odd -- 2(2m+1)
- odd-even --
Similar subdivisions of odd numbers are:
- prime and incomposite -- ordinary primes excluding 2,
- secondary and composite -- ordinary composite with prime factors only,
- 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 is prime, then 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-- . So, 12 is abundant.
Example: All primes are deficient.
Just the fact of finding perfect numbers using the previous propositions has spawned a cottage industry of determining those numbers p for which is prime. Such primes are called Mersenne (1588-1648) primes after the friar of the 17 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 Primesnumber 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.
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 . The next number is 2n+1, which we suppose to be a square.
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.
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
Contributions by the Pythagorean s include
- Various theorems about triangles, parallel lines, polygons, circles, spheres and regular polyhedra.
- Work on a class of problems in the applications of areas. (e.g. to construct a polygon of given area and similar to another polygon.)
- Given a line segment, construct on part of it or on the line segment extended a parallelogram equal to a given rectilinear figure in area and falling short or exceeding by a parallelogram similar to a given one. (In modern terms, solve .)
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 section 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 accepted, 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.
The key to the compass and ruler construction of the pentagon is the construction of the isoceles triangle with angles and . 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 PAB and QPB. Then . This implies , and hence . Solving for we, get . Since PBQ is isoceles, the angle QBP . 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.
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 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.
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 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 ( 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. is incommensurable with 1.
Proof. Suppose that , with no common factors. Then
Thus , and hence . So, a=2c and it follows that
whence by the same reasoning yields that . 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." 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. 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).
The large lune ABD is similar to the smaller lune (with base on one leg of the right isosceles triangle . Proposition:The area of the large lune ABD is the area of the semi-circle less the area of the triangle .
Next: About this document Don Allen
Thu Feb 6 13:10:23 CST 1997
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