Pythagorean theorem

[Justin-sama]

Abu Jafar al-Khazin may have worked on both astronomy and number theory or there may have been two mathematicians both working around the same period, one working on astronomy and one on number theory. As far as this article is concerned we will assume that al-Khazin worked on both topics. There seems no way of being certain which position is correct.

Al-Khazin's family were from Saba, a kingdom in southwestern Arabia, perhaps better known as Sheba from the biblical story of King Solomon and the Queen of Sheba. In the Fihrist, a tenth century survey of Islamic culture, he is described Al-Khurasani which means that he came from Khurasan in eastern Iran.

The Buyid dynasty, ruling in western Iran and Iraq, reach its peak around the time that al-Khazin lived. It undertook public schemes such as building hospitals and dams, as well as patronising the arts and sciences. Rayy, situated southeast of present day Tehran, was one of the major cultural centres of the Buyid dynasty. Islamic writers described Rayy as:-

... a city of extraordinary beauty, built largely of fired brick and brilliantly ornamented with blue faience (glazed earthenware).

Al-Khazin was one of the scientists brought to the court in Rayy by the ruler of the Buyid dynasty, Adud ad-Dawlah, who ruled from 949 to 983. We know that in 959/960 al-Khazin was required by the vizier of Rayy, who was appointed by Adud ad-Dawlah, to measure the obliquity of the ecliptic (the angle which the plane in which the sun appears to move makes with the equator of the earth). He is said to have made the measurement:-

... using a ring of about 4 meters.

One of al-Khazin's works Zij al-Safa'ih (Tables of the disks of the astrolabe) was described by his successors as the best work in the field and they make many reference to it. The work describes some astronomical instruments, in particular it describes an astrolabe fitted with plates inscribed with tables and a commentary on the use of these. A copy of this instrument was made but vanished in Germany at the time of World War II. A photograph of this copy was taken and the article [5] examines this.

Al-Khazin wrote a commentary on Ptolemy's Almagest which was criticised by al-Biruni for being too verbose. Only one fragment of this commentary has survived and a translation of it is given in [6]. The fragment which has survived contains a discussion by al-Khazin of Ptolemy's argument that the universe is spherical. Ptolemy wrote [6]:-

.. of different figures of equal perimeter, the one with more angles is greater in capacity, and therefore it is necessary that a circle is the greatest of surfaces (i.e. of all plane figures with a constant perimeter) and the sphere the greatest of solids.

Al-Khazin gives 19 propositions relating to this statement by Ptolemy. The most interesting results show, with a very ingenious proof, that an equilateral triangle has a greater area than any isosceles or scalene triangle with the same perimeter. When he tries to generalise this result to polygons, however, al-Khazin gives incorrect proofs. Other results among the 19 are based on propositions given by Archimedes in On the sphere and cylinder. The author of [6] argues that the ingenious results on triangles are unlikely to be due to al-Khazin but are probably taken by him from some unknown source.

The suggestion in [6] that al-Khazin is a third rate mathematician is somewhat doubtful given his work on number theory but as we stated at the beginning of this article, it is possible that there were two mathematicians of the same name. The papers [4], [9] and [7] all look at this number theory work by al-Khazin (see also [2] and [3]). The work of al-Khazin which is described seems to have been motivated by work of a mathematician by the name of al-Khujandi.

Al-Khujandi claimed to have proved that x3 + y3 = z3 is impossible for whole numbers x, y, z which of course is the n = 3 case of Fermat's Last Theorem. In a letter al-Khazin wrote:-

I demonstrate earlier ... that what Abu Muhammad al-Khujandi advanced - may God have mercy on him - in his demonstration that the sum of two cubic numbers is not a cube is defective and incorrect.

This seems to have motivated further correspondence on number theory between al-Khazin and other Arabic mathematicians. Results by al-Khazin here are interesting indeed. His main result is to:-

... show how, if we are given a number, to find a square number so that if the given number were added to it or subtracted from it the result would be square.

In modern notation the problem is given a natural number a, find natural numbers x, y, z so that x2 + a = y2 and x2 - a = z2. Al-Khazin proves that the existence of x, y, z with these properties is equivalent to the existence of natural numbers u, v with a = 2uv, and u2 + v2 is a square (in fact u2 + v2 = x2). The smallest example of a satisfying these conditions is 24 which al-Khazin gives

52 + 24 = 72, 52 - 24 = 12.

He also gives a = 96 with

102 + 96 = 142, 102 - 96 = 22

although, rather strangely, he seems to discount this case by another of his statements. Rashed suggests this may be because 96 = 2 48 = 2 6 8 and 62 + 82 = 102is not a primitive Pythagorean triple.

There is a mystery which Rashed notes in [7] (also in [2] and [3]). This relates to the quote above by al-Khazin regarding the false proof by al-Khujandi of the impossibility of proving x3 + y3 = z3. Rashed has discovered a manuscript which appears to be by al-Khazin, yet contains exactly what he had attributed to al-Khujandi. Although al-Khazin could have realised the error in al-Khujandi's proof and attempted a similar proof himself which he believed correct, there is no really satisfactory explanation of these facts.

Finally we should mention that al-Khazin proposed a different solar model from that of Ptolemy. Ptolemy had the sun moving in uniform circular motion about a centre which was not the earth. Al-Khazin was unhappy with this model since he claimed that if this were the case then the apparent diameter of the sun would vary throughout the year and observation showed that this were not the case. Of course the apparent diameter of the sun does vary but by too small an amount to be observed by al-Khazin. To get round this problem, al-Khazin proposed a model in which the sun moved in a circle which was centred on the earth, but its motion was not uniform about the centre, rather it was uniform about another point (called the excentre).

[Anriasia]

The History of Pythagoras and his Theorem
In this section you will learn about the life of Pythagoras and how it is that the theorem is known as the Pythagorean Theorem.
Be aware that there are no good records about the life of Pythagoras, so the exact dates and other issues are not known with certainty. In addition, the names of some of the people as well as the places where Pythagoras lived may have different spellings.
Pythagoras was born in the island of Samos in ancient Greece1. There is no certainty regarding the exact year when he was born, but it is believed that it was around 570 BC That is about 2,570 years ago! Those were times when a person believed in superstitions and had strong beliefs in gods, spirits, and the mysterious. Religious cults were very popular in those times.
Pythagoras of Samos
Pythagoras' father's name was Mnesarchus and may have been a Phoenician. His mother's name was Pythais. Mnesarchus made sure that his son would get the best possible education. His first teacher was Pherecydes, and Pythagoras stayed in touch with him until Pherecydes' death. When Pythagoras was about 18 years old he went to the island of Lesbos where he worked and learned from Anaximander, an astronomer and philosopher, and Thales of Miletus, a very wise philosopher and mathematician.
Thales had visited Egypt and recommended that Pythagoras go to Egypt. Pythagoras arrived in Egypt around 547 BC when he was 23 years old. He stayed in Egypt for 21 years learning a variety of things including geometry from Egyptian priests . It was probably in Egypt where he learned the theorem that is now called by his name.
By the time he was about 55 years old he returned to his native land and started a school on the island of Samos. However, because of the lack of students he decided to move to Croton in the south of Italy.
In Croton he started a school which concentrated in the teaching and learning of Mathematics, Music, Philosophy, and Astronomy and their relationship with Religion. It is said that as many as 600 of the worthiest people in the city attended the school, including Theana whom he married when he was 60. The school reached its highest splendor around the year 490 BC. He taught the young to respect their elders and to develop their mind through learning. He emphasized justice based on equality. Calmness and gentleness were principles encouraged at the school. Pythagoreans became known for their close friendships and devotion to each other. More than anyone before him Pythagoras combined the spiritual teachings with the pursuit of knowledge and science.
Pythagoras also headed a cult known as the secret brotherhood that worshiped numbers and numerical relationships. They attempted to find mathematical explanations for music, the gods, the cosmos, etc. Pythagoras believed that all relations could be reduced to number relations.
At some point Pythagoras was exiled from Croton and had to move to Tarentum. After 16 years he had to move again, this time to Metapontus where he lived four years before he died at the age of 99.
Here we have a picture of a statue of Phytagoras in the island of Samos. If you click on the figure you'll be able to see a larger picture. On the bottom of the statue the text is "". The literal translation is "Pythagoras the Samosan", but the preferred translation is "Pythagoras of Samos".
Now let's talk a bit about the theorem that bears his name. The Egyptians knew that a triangle with sides 3, 4, and 5 make a 90o angle. As a matter of fact, they had a rope with 12 evenly spaced knots.
that they used to build perfect corners in their buildings and pyramids. It is believed that they only knew about the 3, 4, 5 triangle and not the general theorem that applies to all right triangles.
The Chinese also knew this theorem. It is attributed to Tschou-Gun who lived in 1100 BC. He knew the characteristics of the right angle. The theorem was also known to the Caldeans and the Babylonians more than a thousand years before Pythagoras. A clay tablet of Babylonian origin was found with the following inscription: "4 is the length and 5 the diagonal. What is the breadth?"
So why is it called the Pythagorean Theorem? Even though the theorem was known long before his time, Pythagoras certainly generalized it and made it popular. It was Pythagoras who is attributed with its first geometrical demonstration. That is why it is known as the Pythagorean Theorem. There are hundreds of purely geometric demonstrations as well as an unlimited (that is right -- an infinite number) of algebraic proofs.
The Pythagorean Theorem is one of the most important theorems in the whole realm of geometry. We will conclude this section by stating the theorem in words:
The square described upon the hypotenuse of a right-angled triangle is equal to the sum of the squares described upon the other two sides.
Another way of saying the same thing is:
When the two shorter sides in a right triangle are squared and then added, the sum equals the square of the longest side or hypotenuse.
You can now move on to the last section and work on some interesting problems.

[Justin-sama]

Simon Stevin's father was Anthuenis (Anton) Stevin who, it is believed, was a cadet son of a mayor of Veurne. His mother was Cathelijne (or Catelyne) van der Poort who was the daughter of a burgher family of Ypres. Anthuenis and Cathelijne were not married but Simon's mother Cathelijne later married a man who was involved in selling carpets and in the silk trade. By marriage Cathelijne joined a family who were Calvinists. Nothing is known of Simon's early years or of his education although one assumes he was brought up in the Calvinist tradition.

Stevin became a bookkeeper and cashier with a firm in Antwerp. It is known that he spent some time between the years 1571 to 1577 travelling in Poland, Prussia and Norway. Then in 1577 he took a job as a clerk in the tax office at Brugge. After this he moved to Leiden in 1581 where he first attended the Latin school, then he entered the University of Leiden in 1583 (at the age of 35). Various theories have been put forward as to why he moved to Leiden. To understand these we need to look briefly at the history of the period.

The Union of Utrecht on 23 January 1579 was designed to form a block (known as the States-General) within the larger union of the Low Countries which would resist Spanish rule. It produced a union in the north Netherlands, still officially under the rule of the King of Spain, but distinct from the south. The strong reaction against the Spanish followed the start of a reign of terror by the Spanish occupation in the south beginning around 1567. The north was predominantly Calvinist and effectively ruled by William, Prince of Orange. In 1581 the States-General declared independence from Spain and a complex situation followed as foreign help was enlisted.

Stevin's move to the north Netherlands certainly coincided with their move to independence from the King of Spain. There were other possible reasons for Stevin to move, however, for we have already mentioned that Stevin was brought up in a Calvinist family after his mother remarried. Certainly Stevin was not alone in fleeing from the south Netherlands around this time, with many going to the north, but others fleeing to England or Germany. While Stevin was at the University of Leiden he met Maurits (Maurice), the Count Of Nassau, who was William of Orange's second son. The two became close friends and Stevin became mathematics tutor to the Prince as well as a close advisor. William of Orange was assassinated on 10 July 1584 at Delft by a Roman Catholic who believed that by assassinating William he would prevent the rebellion against Catholic Spain. William's eldest son Philip William was loyal to Spain so it was Maurits who was appointed stadholder of Holland and Zeeland, or the United Provinces of the Netherlands, in 1584.

With Prince Maurits now head of the army of the republic, and with Stevin as an advisor in his service, a series of military triumphs over the Spanish forces followed. Maurits understood the importance of military strategy, tactics, and engineering in military success. In 1600 he asked Stevin to set up an engineering school within the University of Leiden. It was a good political move to insist that the courses were taught there in the Dutch language. Certainly Prince Maurits saw his friend Stevin as having major importance in his success and the recent discovery of a journal in the Public Record Office of The Hague recording Stevin's salary as 600 Dutch guilders in 1604 confirms his high position.

It is believed that from 1604 Stevin was quartermaster-general of the army of the States-General. He invented a way of flooding the lowlands in the path of an invading army by opening selected sluices in dikes. He was an outstanding engineer who advised on building windmills, locks and ports. He advised Prince Maurits on building fortifications for the war against Spain and wrote detailed descriptions of the military innovations adopted by the army. These innovations would be copied by many other countries.

The army of the States-General reclaimed from Spanish rule essentially the territory which is today The Netherlands, and the States-General became officially recognized by England and France as an independent state. Prince Maurits wished to continue the war against Spain but, when Spain effectively recognised the United Provinces as independent and sovereign, there was little enthusiasm to continue the fight. The Twelve Years' Truce began in 1609.

Stevin bought a house at the Raamstraat in The Hague in 1612 for 3800 Dutch guilders (another sign of his high status and wealth). He married at a date given as 1610 by some sources and as 1614 by other sources. His wife was Catherine Krai, and they had four children named Frederic, Hendrik, Susanna and Levina. Hendrik, their second child, went on to attend the University of Leiden and, becoming a famous scientist in his own right, was the editor of his father's collected works.

The author of 11 books, Simon Stevin made significant contributions to trigonometry, mechanics, architecture, musical theory, geography, fortification, and navigation. His first book was Tafelen van Interest (Tables of interest) which he published in 1582. Prior to this, unpublished manuscript interest tables were in common use with bankers throughout Europe but had been treated as secret information not to be divulged. Before presenting the numerical tables, Stevin gave rules for simple and compound interest and also gave many examples of their use.

In Problemata geometrica (1583) Stevin presented geometry based largely on Euclid and Archimedes but the problems which he studied show that he was also influenced by D¨¹rer. Stevin gave an interesting account in this work of constructions related to polygons and polyhedra, using the concept of similarity, and a study of regular and semi-regular polyhedra. It was written in Latin, and is the only one of his books to be first published in that language. He became a strong advocate of writing his scientific works in Dutch and he gives clear reasons for this choice in a text written in 1586.

In 1585 he published La Theinde (The tenth), a twenty-nine page booklet in which he presented an elementary and thorough account of decimal fractions. He wrote this small book for the benefit of:-

... stargazers, surveyors, carpet-makers, wine-gaugers, mint-masters and all kind of merchants.

Although he did not invent decimals (they had been used by the Arabs and the Chinese long before Stevin's time) he did introduce their use in mathematics in Europe. Stevin states that the universal introduction of decimal coinage, measures and weights would only be a matter of time (but he probably would be amazed to know that in the 21st century some countries still resist adopting decimal systems). Robert Norton published an English translation of La Theinde in London in 1608. It was titled Disme, The Arts of Tenths or Decimal Arithmetike and it was this translation which inspired Thomas Jefferson to propose a decimal currency for the United States (note that one tenth of a dollar is still called a dime). Stevin's notation was to be taken up by Clavius and Napier and it developed into that used today.

In the same year (1585) he published La pratique d'arithm¨¦tique and L'arithm¨¦tique which were the only texts he wrote first in French. In the latter Stevin presented a unified treatment for solving quadratic equations and a method for finding approximate solutions to algebraic equations of all degrees. He also made a strong plea that all numbers such as square roots, irrational numbers, surds, negative numbers etc should all be treated as numbers and not distinguished as being different in nature. Stevin's notion of a real number was accepted by essentially all later scientists. Particularly important was Stevin's acceptance of negative numbers but he did not accept the 'new' imaginary numbers and this was to hold back their development.

Inspired by Archimedes, Stevin wrote important works on mechanics. Mainly dealing with statics, his treatment appears in his book De Beghinselen der Weegconst published in 1586. It is famous for containing the theorem of the triangle of forces which gave impetus to statics. In the same year his treatise De Beghinselen des Waterwichts on hydrostatics contained notable improvements to the work of Archimedes on this topic. Many consider that he founded the science of hydrostatics with this work by showing that the pressure exerted by a liquid upon a given surface depends on the height of the liquid and the area of the surface.

Also in 1586 (3 years before Galileo) he reported that different weights fell a given distance in the same time. His experiments were conducted using two lead balls, one being ten times the weight of the other, which he dropped thirty feet from the church tower in Delft.

In De Hemelloop, published in 1608, he wrote on astronomy and strongly defended the sun centred system of Copernicus. Although he undertook his mathematical work earlier in his life, Stevin collected together some of his mathematical writings which he edited and published during the years 1605 to 1608 in Wiskonstighe Ghedachtenissen (Mathematical Memoirs). The collection included De Driehouckhandel (Trigonometry), De Meetdaet (Practice of measuring), and De Deursichtighe (Perspective). The work on perspective looks at a number of innovations such as the case of calculating the perspective for making a drawing on a canvas which is not perpendicular to the ground, and the case of inverse perspective. This calculates where the eye of the observer should be placed if an object and a perspective drawing of that object are given. Stevin, in his book Stelreghel (Algebra) used the notation +, - and ¡Ì.

His other works included Vita Politica. Het Burgherlick leven (Civil life) published in 1590, De Sterktenbouwing (The construction of fortifications) published in 1594, De Havenvinding (Position finding) published in 1599, and the double work Castrametatio, dat is legermeting and Nieuwe Maniere van Stercktebou door Spilsluysen published in 1617.

In Het Burgherlick leven Stevin discusses how a citizen of a state should comply with the rules of the authorities (even when they appear unjust) and, in particular, he advises citizens how to behave in times of civil unrest. In De Sterktenbouwing Stevin takes an Italian method of fortification and modifies it for Dutch use. The ideas that he put forward in this treatise were clever but too expensive to implement. The work De Havenvinding literally means 'finding the harbour' and presents a method of finding the position of a ship by determining its longitude using the magnetic variation of the compass needle. Although theoretically sound, the method is impractical. In the first of the final double work that we mentioned above, Stevin describes the establishment, layout and setting up of a military camp. Particularly fascinating is his description of Prince Maurits camp which he set up prior to the Battle of Juliers in 1610. The second of the two works deals with sluices Stevin had designed to put into fortifications to keep a moat at the correct depth.

His contributions to music are contained in De Spiegheling der Singconst (Theory of the art of singing) which survived in manuscript until 1884 when it was published. This is usually seen as the first correct theory of the division of the octave into twelve equal intervals, see for example [1]. Cohen in [13] explains the importance of the problem to scientists of the period:-

Many pioneers of the Scientific Revolution, such as Galileo, Kepler, Stevin, Descartes, Mersenne, and others, wrote extensively about music theory. This was not a chance interest of a few individual scientists. Rather, it reflects a continuing concern of scientists from Pythagorean times onwards to solve certain quantifiable problems in music theory. One of the issues involved was technically known as 'the division of the octave', the problem, that is, with which notes to make music.

Cohen argues that this was not, as is commonly believed (see [1]), the purpose of Stevin's treatise:-

A careful analysis of the problem situation in the science of music around 1600, reveals that Stevin's treatise highlights a particular stage in the history of what has always been the core issue of the science of music, namely, the problem of consonance. This is the search for an explanation, on scientific principles, of Pythagoras' law: Why is it that those few musical intervals which affect our ear in a sweet and pleasing manner, correspond to the ratios of the first few integers?.

[Anriasia]

A Brief History of the Pythagorean Theorem


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Just Who Was This Pythagoras, Anyway?
Pythagoras (569-500 B.C.E.) was born on the island of Samos in Greece, and did much traveling through Egypt, learning, among other things, mathematics. Not much more is known of his early years. Pythagoras gained his famous status by founding a group, the Brotherhood of Pythagoreans, which was devoted to the study of mathematics. The group was almost cult-like in that it had symbols, rituals and prayers. In addition, Pythagoras believed that "Number rules the universe,"and the Pythagoreans gave numerical values to many objects and ideas. These numerical values, in turn, were endowed with mystical and spiritual qualities.

Legend has it that upon completion of his famous theorem, Pythagoras sacrificed 100 oxen. Although he is credited with the discovery of the famous theorem, it is not possible to tell if Pythagoras is the actual author. The Pythagoreans wrote many geometric proofs, but it is difficult to ascertain who proved what, as the group wanted to keep their findings secret. Unfortunately, this vow of secrecy prevented an important mathematical idea from being made public. The Pythagoreans had discovered irrational numbers! If we take an isosceles right triangle with legs of measure 1, the hypotenuse will measure sqrt 2. But this number cannot be expressed as a length that can be measured with a ruler divided into fractional parts, and that deeply disturbed the Pythagoreans, who believed that "All is number." They called these numbers "alogon," which means "unutterable." So shocked were the Pythagoreans by these numbers, they put to death a member who dared to mention their existence to the public. It would be 200 years later that the Greek mathematician Eudoxus developed a way to deal with these unutterable numbers.



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The sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse.
This relationship has been known since the days of the ancient Babylonians and Egyptians, although it may not have been stated as explicitly as above. A portion of a 4000 year old Babylonian tablet (c. 1900 B.C.E.), now known as Plimpton 322, (in the collection of Columbia University, New York), lists columns of numbers showing what we now call Pythagorean Triples--sets of numbers that satisfy the equation


a^2 + b^2 = c^2

[Anriasia]

I. The Pythagorean Theorem

To begin, the Pythagorean theorem states that the square on the hypotenuse of a right triangle has an area equal to the combined areas of the squares on the other two sides. (Gardner, 152) One will find the converse of this statement to also be true.

The Pythagorean theorem was a mathematical fact that the Babylonians knew and used. However 1000 years later, between the years of 580-500 BC, Pythagoras of Samos was the first to prove the theorem. It is possible that someone proved the theorem before Pythagoras, but no proof has been found. Because of this, Pythagoras is given credit for the first proof. (MacTutor History of Mathematics Archive)

Before a proof was ever given, besides the Babylonians it was thought that "Egyptian temple builders used ropes in laying foundations, suggested that perhaps the obtained accurate right angles by using marked ropes that could be stretched around stakes to form a 3,4,5 right triangle." There is no documented evidence of this but Cantor, a historian of mathematics agreed this could be true. (Gardner, 155)

Following the first proof, many proofs followed. Proofs have been found by Euclid, Socrates, and even President Garfield. The Pythagorean Proposition, by Elisha S. Loomis gives 367 different proofs of this theorem. With the Pythagorean theorem being such a popular topic, it is no wonder high school students study the theorem. (Gardner)

Before the proofs, it is important for students to see the theorem as it is worded. "The square on the hypotenuse of a right triangle has an area equal to the combined areas of the squares on the other two sides" can be drawn to get a better idea about what this proof is stating.


I+II=III where I, II, and III represent areas of the squares.

This definition gives rise to the definition that is used in today's classroom textbooks. "In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse." (Algebra, 414)



a2`+b2=c2

Two proofs of the Pythagorean theorem are available from this page and both of them are suitable to teach in the classroom. One proof may predate Pythagoras, and the second proof President Garfield devised.

Proofs of the Pythagorean Theorem

In order to fully understand the Pythagorean theorem, it is important to know that it is only a generalization of the law of cosines.

Law of Cosines

Lastly, here are a few exercises to play with:

Exercises

[Anriasia]

A Biography!!! Woo hoo! xD
Pythagoras of Samos is often described as the first pure mathematician. He is an extremely important figure in the development of mathematics yet we know relatively little about his mathematical achievements. Unlike many later Greek mathematicians, where at least we have some of the books which they wrote, we have nothing of Pythagoras's writings. The society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure.

We do have details of Pythagoras's life from early biographies which use important original sources yet are written by authors who attribute divine powers to him, and whose aim was to present him as a god-like figure. What we present below is an attempt to collect together the most reliable sources to reconstruct an account of Pythagoras's life. There is fairly good agreement on the main events of his life but most of the dates are disputed with different scholars giving dates which differ by 20 years. Some historians treat all this information as merely legends but, even if the reader treats it in this way, being such an early record it is of historical importance.

Pythagoras's father was Mnesarchus ([12] and [13]), while his mother was Pythais [8] and she was a native of Samos. Mnesarchus was a merchant who came from Tyre, and there is a story ([12] and [13]) that he brought corn to Samos at a time of famine and was granted citizenship of Samos as a mark of gratitude. As a child Pythagoras spent his early years in Samos but travelled widely with his father. There are accounts of Mnesarchus returning to Tyre with Pythagoras and that he was taught there by the Chaldaeans and the learned men of Syria. It seems that he also visited Italy with his father.

Little is known of Pythagoras's childhood. All accounts of his physical appearance are likely to be fictitious except the description of a striking birthmark which Pythagoras had on his thigh. It is probable that he had two brothers although some sources say that he had three. Certainly he was well educated, learning to play the lyre, learning poetry and to recite Homer. There were, among his teachers, three philosophers who were to influence Pythagoras while he was a young man. One of the most important was Pherekydes who many describe as the teacher of Pythagoras.

The other two philosophers who were to influence Pythagoras, and to introduce him to mathematical ideas, were Thales and his pupil Anaximander who both lived on Miletus. In [8] it is said that Pythagoras visited Thales in Miletus when he was between 18 and 20 years old. By this time Thales was an old man and, although he created a strong impression on Pythagoras, he probably did not teach him a great deal. However he did contribute to Pythagoras's interest in mathematics and astronomy, and advised him to travel to Egypt to learn more of these subjects. Thales's pupil, Anaximander, lectured on Miletus and Pythagoras attended these lectures. Anaximander certainly was interested in geometry and cosmology and many of his ideas would influence Pythagoras's own views.

In about 535 BC Pythagoras went to Egypt. This happened a few years after the tyrant Polycrates seized control of the city of Samos. There is some evidence to suggest that Pythagoras and Polycrates were friendly at first and it is claimed [5] that Pythagoras went to Egypt with a letter of introduction written by Polycrates. In fact Polycrates had an alliance with Egypt and there were therefore strong links between Samos and Egypt at this time. The accounts of Pythagoras's time in Egypt suggest that he visited many of the temples and took part in many discussions with the priests. According to Porphyry ([12] and [13]) Pythagoras was refused admission to all the temples except the one at Diospolis where he was accepted into the priesthood after completing the rites necessary for admission.

It is not difficult to relate many of Pythagoras's beliefs, ones he would later impose on the society that he set up in Italy, to the customs that he came across in Egypt. For example the secrecy of the Egyptian priests, their refusal to eat beans, their refusal to wear even cloths made from animal skins, and their striving for purity were all customs that Pythagoras would later adopt. Porphyry in [12] and [13] says that Pythagoras learnt geometry from the Egyptians but it is likely that he was already acquainted with geometry, certainly after teachings from Thales and Anaximander.

In 525 BC Cambyses II, the king of Persia, invaded Egypt. Polycrates abandoned his alliance with Egypt and sent 40 ships to join the Persian fleet against the Egyptians. After Cambyses had won the Battle of Pelusium in the Nile Delta and had captured Heliopolis and Memphis, Egyptian resistance collapsed. Pythagoras was taken prisoner and taken to Babylon. Iamblichus writes that Pythagoras (see [8]):-

... was transported by the followers of Cambyses as a prisoner of war. Whilst he was there he gladly associated with the Magoi ... and was instructed in their sacred rites and learnt about a very mystical worship of the gods. He also reached the acme of perfection in arithmetic and music and the other mathematical sciences taught by the Babylonians...

In about 520 BC Pythagoras left Babylon and returned to Samos. Polycrates had been killed in about 522 BC and Cambyses died in the summer of 522 BC, either by committing suicide or as the result of an accident. The deaths of these rulers may have been a factor in Pythagoras's return to Samos but it is nowhere explained how Pythagoras obtained his freedom. Darius of Persia had taken control of Samos after Polycrates' death and he would have controlled the island on Pythagoras's return. This conflicts with the accounts of Porphyry and Diogenes Laertius who state that Polycrates was still in control of Samos when Pythagoras returned there.

Pythagoras made a journey to Crete shortly after his return to Samos to study the system of laws there. Back in Samos he founded a school which was called the semicircle. Iamblichus [8] writes in the third century AD that:-

... he formed a school in the city [of Samos], the 'semicircle' of Pythagoras, which is known by that name even today, in which the Samians hold political meetings. They do this because they think one should discuss questions about goodness, justice and expediency in this place which was founded by the man who made all these subjects his business. Outside the city he made a cave the private site of his own philosophical teaching, spending most of the night and daytime there and doing research into the uses of mathematics...

Pythagoras left Samos and went to southern Italy in about 518 BC (some say much earlier). Iamblichus [8] gives some reasons for him leaving. First he comments on the Samian response to his teaching methods:-

... he tried to use his symbolic method of teaching which was similar in all respects to the lessons he had learnt in Egypt. The Samians were not very keen on this method and treated him in a rude and improper manner.

This was, according to Iamblichus, used in part as an excuse for Pythagoras to leave Samos:-

... Pythagoras was dragged into all sorts of diplomatic missions by his fellow citizens and forced to participate in public affairs. ... He knew that all the philosophers before him had ended their days on foreign soil so he decided to escape all political responsibility, alleging as his excuse, according to some sources, the contempt the Samians had for his teaching method.

Pythagoras founded a philosophical and religious school in Croton (now Crotone, on the east of the heel of southern Italy) that had many followers. Pythagoras was the head of the society with an inner circle of followers known as mathematikoi. The mathematikoi lived permanently with the Society, had no personal possessions and were vegetarians. They were taught by Pythagoras himself and obeyed strict rules. The beliefs that Pythagoras held were [2]:-

(1) that at its deepest level, reality is mathematical in nature,
(2) that philosophy can be used for spiritual purification,
(3) that the soul can rise to union with the divine,
(4) that certain symbols have a mystical significance, and
(5) that all brothers of the order should observe strict loyalty and secrecy.

Both men and women were permitted to become members of the Society, in fact several later women Pythagoreans became famous philosophers. The outer circle of the Society were known as the akousmatics and they lived in their own houses, only coming to the Society during the day. They were allowed their own possessions and were not required to be vegetarians.

Of Pythagoras's actual work nothing is known. His school practised secrecy and communalism making it hard to distinguish between the work of Pythagoras and that of his followers. Certainly his school made outstanding contributions to mathematics, and it is possible to be fairly certain about some of Pythagoras's mathematical contributions. First we should be clear in what sense Pythagoras and the mathematikoi were studying mathematics. They were not acting as a mathematics research group does in a modern university or other institution. There were no 'open problems' for them to solve, and they were not in any sense interested in trying to formulate or solve mathematical problems.

Rather Pythagoras was interested in the principles of mathematics, the concept of number, the concept of a triangle or other mathematical figure and the abstract idea of a proof. As Brumbaugh writes in [3]:-

It is hard for us today, familiar as we are with pure mathematical abstraction and with the mental act of generalisation, to appreciate the originality of this Pythagorean contribution.

In fact today we have become so mathematically sophisticated that we fail even to recognise 2 as an abstract quantity. There is a remarkable step from 2 ships + 2 ships = 4 ships, to the abstract result 2 + 2 = 4, which applies not only to ships but to pens, people, houses etc. There is another step to see that the abstract notion of 2 is itself a thing, in some sense every bit as real as a ship or a house.

Pythagoras believed that all relations could be reduced to number relations. As Aristotle wrote:-

The Pythagorean ... having been brought up in the study of mathematics, thought that things are numbers ... and that the whole cosmos is a scale and a number.

This generalisation stemmed from Pythagoras's observations in music, mathematics and astronomy. Pythagoras noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and that these ratios could be extended to other instruments. In fact Pythagoras made remarkable contributions to the mathematical theory of music. He was a fine musician, playing the lyre, and he used music as a means to help those who were ill.

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 recognise as mathematics today [3]:-

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.

Of course 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. Proclus, the last major Greek philosopher, who lived around 450 AD wrote (see [7]):-

After [Thales, etc.] Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner: he it was who discovered the theory of irrational and the construction of the cosmic figures.

Again Proclus, writing of geometry, said:-

I emulate the Pythagoreans who even had a conventional phrase to express what I mean "a figure and a platform, not a figure and a sixpence", by which they implied that the geometry which is deserving of study is that which, at each new theorem, sets up a platform to ascend by, and lifts the soul on high instead of allowing it to go down among the sensible objects and so become subservient to the common needs of this mortal life.

Heath [7] gives 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 generalisation 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 centre of the Universe. He also recognised that the orbit of the Moon was inclined to the equator of the Earth and he was one of the first to realise that Venus as an evening star was the same planet as Venus as a morning star.

Primarily, however, Pythagoras was a philosopher. In addition to his beliefs about numbers, geometry and astronomy described above, he held [2]:-

... the following philosophical and ethical teachings: ... the dependence of the dynamics of world structure on the interaction of contraries, or pairs of opposites; the viewing of the soul as a self-moving number experiencing a form of metempsychosis, or successive reincarnation in different species until its eventual purification (particularly through the intellectual life of the ethically rigorous Pythagoreans); and the understanding ...that all existing objects were fundamentally composed of form and not of material substance. Further Pythagorean doctrine ... identified the brain as the locus of the soul; and prescribed certain secret cultic practices.

In [3] their practical ethics are also described:-

In their ethical practices, the Pythagorean were famous for their mutual friendship, unselfishness, and honesty.

Pythagoras's Society at Croton was not unaffected by political events despite his desire to stay out of politics. Pythagoras went to Delos in 513 BC to nurse his old teacher Pherekydes who was dying. He remained there for a few months until the death of his friend and teacher and then returned to Croton. In 510 BC Croton attacked and defeated its neighbour Sybaris and there is certainly some suggestions that Pythagoras became involved in the dispute. Then in around 508 BC the Pythagorean Society at Croton was attacked by Cylon, a noble from Croton itself. Pythagoras escaped to Metapontium and the most authors say he died there, some claiming that he committed suicide because of the attack on his Society. Iamblichus in [8] quotes one version of events:-

Cylon, a Crotoniate and leading citizen by birth, fame and riches, but otherwise a difficult, violent, disturbing and tyrannically disposed man, eagerly desired to participate in the Pythagorean way of life. He approached Pythagoras, then an old man, but was rejected because of the character defects just described. When this happened Cylon and his friends vowed to make a strong attack on Pythagoras and his followers. Thus a powerfully aggressive zeal activated Cylon and his followers to persecute the Pythagoreans to the very last man. Because of this Pythagoras left for Metapontium and there is said to have ended his days.

This seems accepted by most but Iamblichus himself does not accept this version and argues that the attack by Cylon was a minor affair and that Pythagoras returned to Croton. Certainly the Pythagorean Society thrived for many years after this and spread from Croton to many other Italian cities. Gorman [6] argues that this is a strong reason to believe that Pythagoras returned to Croton and quotes other evidence such as the widely reported age of Pythagoras as around 100 at the time of his death and the fact that many sources say that Pythagoras taught Empedokles to claim that he must have lived well after 480 BC.

The evidence is unclear as to when and where the death of Pythagoras occurred. Certainly the Pythagorean Society expanded rapidly after 500 BC, became political in nature and also spilt into a number of factions. In 460 BC the Society [2]:-

... was violently suppressed. Its meeting houses were everywhere sacked and burned; mention is made in particular of "the house of Milo" in Croton, where 50 or 60 Pythagoreans were surprised and slain. Those who survived took refuge at Thebes and other places.

[Psychoness]

Not to interupt but, wow. Who knew Pythagoras had such an exciting life! xD

It's so very long and hurts my eyes! >< '

[Anriasia]

It's for a geometry project xD

justin is doing , as you can guess, the Pythagorean theorem, and I am doing Nine Point Circles ^^

[Justin-sama]

Who woulda thought that Pythagoras got to see some Lesbians on the Island of Lesbos

[Anriasia]

I know, I mean, it's just so STRANGE!!!!!

my forehead still burns xD

[Psychoness]

xD That's interesting.

Why does your forehead burn?

[Anriasia]

Oh, I had a fever that day ^^

[Psychoness]

OH POOR SAMMY!! I SHOULD BE THERE!! xP!! NOOO!


YOu okay? 0.o!

[Anriasia]

Lmao, yeah.... I'm fine xD

but everyone else thought I was insane...
i didn't feel very good, so I no longer cared, and wasn't thinking before I say things, and you know how bad I get when I forget to think before speaking... Needless to say, i rather obnoxious...


heff was like, "Holy shit... You need to get sick again, Sam, your so much funnier"

xD yeah...