Pythagoras of Samos - Revision history

Ethomas47: /* Mathematical Contributions */

2015-12-05T00:16:40Z

Mathematical Contributions

← Older revision		Revision as of 20:16, 4 December 2015
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	==Mathematical Contributions==		==Mathematical Contributions==
	Pythagoras's most famous contribution is the formula to find the hypotenuse of a right triangle, known as the Pythagorean Theorem: a^2+b^2=c^2		Pythagoras's most famous contribution is the formula to find the hypotenuse of a right triangle, known as the Pythagorean Theorem: a^2+b^2=c^2

	Heath [1] gives a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans.		Heath [1] gives a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans.

Ethomas47: /* Mathematical Contributions */

2015-12-05T00:16:28Z

Mathematical Contributions

← Older revision		Revision as of 20:16, 4 December 2015
Line 27:		Line 27:

	==Mathematical Contributions==		==Mathematical Contributions==
	Pythagoras's most famous contribution is the formula to find the hypotenuse of a right triangle, known as the Pythagorean Theorem.		Pythagoras's most famous contribution is the formula to find the hypotenuse of a right triangle, known as the Pythagorean Theorem: a^2+b^2=c^2
	Heath [1] gives a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans.		Heath [1] gives a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans.

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	''(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.		''(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.
	''''		''''

	==Connectedness==		==Connectedness==
	Pythagoras is relevant to this course because his mathematical contributions, most importantly the Pythagorean Theorem. The Pythagorean Theorem is essential for geometry and trigonometry, which are used in physics to calculate vectors and angles.		Pythagoras is relevant to this course because his mathematical contributions, most importantly the Pythagorean Theorem. The Pythagorean Theorem is essential for geometry and trigonometry, which are used in physics to calculate vectors and angles.

Ethomas47 at 00:15, 5 December 2015

2015-12-05T00:15:03Z

← Older revision		Revision as of 20:15, 4 December 2015
Line 4:		Line 4:

	Pythagoras of Samos is considered by many to be the first pure mathematician, as well as a Greek philosopher and scientist. He has played an incremental role in the development of mathematics, yet we have none of his written work. Most of what we know about him today was written after his death, and many of the writings regard him as a god-like figure. This leaves very little reliable information on this mystic mathematician.		Pythagoras of Samos is considered by many to be the first pure mathematician, as well as a Greek philosopher and scientist. He has played an incremental role in the development of mathematics, yet we have none of his written work. Most of what we know about him today was written after his death, and many of the writings regard him as a god-like figure. This leaves very little reliable information on this mystic mathematician.

	~~Contents [hide]~~
	~~1 Biography~~
	~~2 Mathematical Contributions~~
	~~3 Connectedness~~
	~~4 External Links~~
	~~5 References~~

	===Early Life===		===Early Life===
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	''(1) that at its deepest level, reality is mathematical in nature,		''(1) that at its deepest level, reality is mathematical in nature,

	(2) that philosophy can be used for spiritual purification,		''(2) that philosophy can be used for spiritual purification,''
			''
	(3) that the soul can rise to union with the divine,		(3) that the soul can rise to union with the divine,''

	(4) that certain symbols have a mystical significance, and		''(4) that certain symbols have a mystical significance, and''

	(5) that all brothers of the order should observe strict loyalty and secrecy.''		''(5) that all brothers of the order should observe strict loyalty and secrecy.''''

	Pythagoras ran his school in great secrecy as well as communalism, so it is difficult to know exactly what they worked on and who produced what.		Pythagoras ran his school in great secrecy as well as communalism, so it is difficult to know exactly what they worked on and who produced what.
Line 39:		Line 32:
	''(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.		''(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.		''(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.		''(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.		''(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.		''(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.		''(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.
	''		''''
	==Connectedness==		==Connectedness==
	Pythagoras is relevant to this course because his mathematical contributions, most importantly the Pythagorean Theorem. The Pythagorean Theorem is essential for geometry and trigonometry, which are used in physics to calculate vectors and angles.		Pythagoras is relevant to this course because his mathematical contributions, most importantly the Pythagorean Theorem. The Pythagorean Theorem is essential for geometry and trigonometry, which are used in physics to calculate vectors and angles.

Ethomas47: by Eleanor Thomas

2015-12-05T00:13:07Z

by Eleanor Thomas

← Older revision		Revision as of 20:13, 4 December 2015
Line 35:		Line 35:
	==Mathematical Contributions==		==Mathematical Contributions==
	Pythagoras's most famous contribution is the formula to find the hypotenuse of a right triangle, known as the Pythagorean Theorem.		Pythagoras's most famous contribution is the formula to find the hypotenuse of a right triangle, known as the Pythagorean Theorem.
	Heath [1] gives a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans.~~[[File:Example.jpg]]~~		Heath [1] 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.		''(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.

Ethomas47 at 00:11, 5 December 2015

2015-12-05T00:11:03Z

← Older revision		Revision as of 20:11, 4 December 2015
Line 1:		Line 1:

	== Pythagoras of Samos ==		== Pythagoras of Samos ==
			by Eleanor Thomas

	Pythagoras of Samos is considered by many to be the first pure mathematician, as well as a Greek philosopher and scientist. He has played an incremental role in the development of mathematics, yet we have none of his written work. Most of what we know about him today was written after his death, and many of the writings regard him as a god-like figure. This leaves very little reliable information on this mystic mathematician.		Pythagoras of Samos is considered by many to be the first pure mathematician, as well as a Greek philosopher and scientist. He has played an incremental role in the development of mathematics, yet we have none of his written work. Most of what we know about him today was written after his death, and many of the writings regard him as a god-like figure. This leaves very little reliable information on this mystic mathematician.
Line 7:		Line 7:
	Contents [hide]		Contents [hide]
	1 Biography		1 Biography
	~~1.1 Early Life~~
	~~1.2 Pythagorean Society~~
	~~1.3 Death~~
	2 Mathematical Contributions		2 Mathematical Contributions
	3 Connectedness		3 Connectedness
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	(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.		(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.
	''		''
	===Connectedness===		==Connectedness==
	Pythagoras is relevant to this course because his mathematical contributions, most importantly the Pythagorean Theorem. The Pythagorean Theorem is essential for geometry and trigonometry, which are used in physics to calculate vectors and angles.		Pythagoras is relevant to this course because his mathematical contributions, most importantly the Pythagorean Theorem. The Pythagorean Theorem is essential for geometry and trigonometry, which are used in physics to calculate vectors and angles.

	===External links===		==External links==
	[1] History of Topics: The Golden Ratio http://www.mathopenref.com/pythagoras.html		[1] History of Topics: The Golden Ratio http://www.mathopenref.com/pythagoras.html

	===References===		==References==
	http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Pythagoras.html		http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Pythagoras.html

	http://www.mathopenref.com/pythagoras.html		http://www.mathopenref.com/pythagoras.html

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2015-12-05T00:08:37Z

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== Pythagoras of Samos ==

Pythagoras of Samos is considered by many to be the first pure mathematician, as well as a Greek philosopher and scientist. He has played an incremental role in the development of mathematics, yet we have none of his written work. Most of what we know about him today was written after his death, and many of the writings regard him as a god-like figure. This leaves very little reliable information on this mystic mathematician.

Contents [hide]
1 Biography
1.1 Early Life
1.2 Pythagorean Society
1.3 Death
2 Mathematical Contributions
3 Connectedness
4 External Links
5 References

===Early Life===
Pythagoras was born around 596 B.C.E. in Samos where he was well-educated. He was taught by Pherekydes, Thales, and his pupil Anaximander. These philosphers introduced Pythagoras to mathematics including geometry and astronomy. After a tyrant seized control of Samos, Pythagoras travelled to Egypt (around 535 BCE). Here he furthered his knowledge in geometry and religion. He was introduced to many customs by Egyptian priests such as refusal to eat meat and beans, refusal to wear animal skin, and the strive for purity. He came to adopt these customs and beliefs later in life. In 525 BCE, the Persians invaded Egypt and Pythagoras was taken prisoner to Babylon. While a prisoner of war, he continued to study mathematics, taught now by the Babylonians. How Pythagoras gained his freedom is still unclear, however the death of the king of Persia most likely played a role. Pythagoras returned to Samos and attempted to teach there. His practices were not very welcome, however, so he left again.

===Pythagorean Society===
Pythagoras left Samos and started a philosophical and religious school in Croton. He was the head of a society with many followers. The inner circle followers were called mathematikoi and lived permanently in the society following many strict rules, including owning no possessions and not eating meat. The outer circle, called the akousmatics, only came to the Society during the day and didn't have as many rules. Women were allowed to be members of the outer circle, many going on to become famous philosophers. At the school, Pythagoras taught mathematics and his beliefs, the main five being:

''(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.''

Pythagoras ran his school in great secrecy as well as communalism, so it is difficult to know exactly what they worked on and who produced what.

===Death===
A noble in Croton attacked the Pythagorean Society because he was denied admittance to the school. Pythagoras fled to Metapontium where most people believed his life ended. Some people believe he returned to the society because it rapidly expanded into 500 BCE.

==Mathematical Contributions==
Pythagoras's most famous contribution is the formula to find the hypotenuse of a right triangle, known as the Pythagorean Theorem.
Heath [1] gives a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans.[[File:Example.jpg]]

''(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.
''
===Connectedness===
Pythagoras is relevant to this course because his mathematical contributions, most importantly the Pythagorean Theorem. The Pythagorean Theorem is essential for geometry and trigonometry, which are used in physics to calculate vectors and angles.

===External links===
[1] History of Topics: The Golden Ratio http://www.mathopenref.com/pythagoras.html

===References===
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Pythagoras.html

http://www.mathopenref.com/pythagoras.html