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PYTHAGORAS
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PYTHAGORAS

Pythagoras of Samos (Ancient GreekΠυθαγόρας ὁ Σάμιος [Πυθαγόρης in Ionian Greek] Pythagóras ho Sámios "Pythagoras the Samian", or simply Πυθαγόρας; b. about 570 – d. about 495 BC) was an Ionian Greek philosophermathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him. He was born on the island of Samos, and might have travelled widely in his youth, visiting Egypt and other places seeking knowledge. Around 530 BC, he moved to Croton, a Greek colony in southern Italy, and there set up a religious sect. His followers pursued the religious rites and practices developed by Pythagoras, and studied his philosophical theories. The society took an active role in the politics of Croton, but this eventually led to their downfall. The Pythagorean meeting-places were burned, and Pythagoras was forced to flee the city.

PYTHAGORAS
PYTHAGORAS   The Pythagoras Theorem   PYTHAGORAS BY RAPHAÈL
imgpythagoras   imgtheorem   imgraphael

 

L I F E

HerodotusIsocrates, and other early writers all agree that Pythagoras was born on Samos, the Greek island in the eastern Aegean, and we also learn that Pythagoras was the son of Mnesarchus. His father was a gem-engraver or a merchant. His name led him to be associated with Pythian Apollo;Aristippus explained his name by saying, "He spoke (agor-) the truth no less than did the Pythian (Pyth-)," and Iamblichus tells the story that the Pythia prophesied that his pregnant mother would give birth to a man supremely beautiful, wise, and beneficial to humankind. A late source gives his mother's name as Pythais. As to the date of his birth, Aristoxenus stated that Pythagoras left Samos in the reign of Polycrates, at the age of 40, which would give a date of birth around 570 BC

Pythagorean theorem

A visual proof of the Pythagorean theorem

Since the fourth century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides—that is, a^2 + b^2 = c^2.

THE PYTHAGOREAN THEOREM
imgtheorem
 

P Y T H A G O R E A N TRIPLE

Pythagorean triple consists of three positive integers ab, and c, such that a2 + b2 = c2. Such a triple is commonly written (abc), and a well-known example is (3, 4, 5). If (abc) is a Pythagorean triple, then so is (kakbkc) for any positive integer k. A primitive Pythagorean triple (PPT)is one in which ab and c are pairwise coprime. A right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle.

The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a2 + b2 = c2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a = b = 1 and c = √2 is right, but (1, 1, √2) is not a Pythagorean triple because √2 is not an integer. Moreover, 1 and √2 do not have an integer common multiple because √2 is irrational.

 

There are 16 primitive Pythagorean triples with c ≤ 100:

( 3 , 4 , 5 ) ( 5, 12, 13) ( 7, 24, 25) ( 8, 15, 17)
( 9, 40, 41) (11, 60, 61) (12, 35, 37) (13, 84, 85)
(16, 63, 65) (20, 21, 29) (28, 45, 53) (33, 56, 65)
(36, 77, 85) (39, 80, 89) (48, 55, 73) (65, 72, 97)

 

Chronological Chart of Sources for Pythagoras

300 CE Iamblichus
(ca. 245–325 CE)
On the Pythagorean Life (extant)
Porphyry
(234–ca. 305 CE)
Life of Pythagoras (extant)
Diogenes Laertius
(ca. 200–250 CE)
Life of Pythagoras (extant)
200 CE Sextus Empiricus
(circa 200 CE)
(summaries of Pythagoras' philosophy in Adversus Mathematicos [Against the Theoreticians], cited below as M.)
100 CE Nicomachus
(ca. 50–150 CE)
Introduction to Arithmetic (extant), Life of Pythagoras (fragments quoted in Iamblichus etc.)
Apollonius of Tyana
(died ca. 97 CE)
Life of Pythagoras (fragments quoted in Iamblichus etc.)
Moderatus of Gades
(50–100)
Lectures on Pythagoreanism (fragments quoted in Porphyry)
Aetius
(first century CE)
Opinions of the Philosophers (reconstructed by H. Diels from pseudo-Plutarch, Opinions of the Philosophers [2nd CE] and Stobaeus,Selections [5th CE])
Pseudo-Pythagorean texts
forged
(starting as early as 300 BCE but most common in the first century BCE)
100 BCE Alexander Polyhistor
(b. 105 BCE)
his excerpts of the Pythagorean Memoirs are quoted by Diogenes Laertius
200 BCE Pythagorean Memoirs
(200 BCE)
(sections quoted in Diogenes Laertius)
300 BCE Timaeus of Tauromenium
350–260 BCE)
(historian of Sicily)
Academy Heraclides
(ca. 380–310)
Xenocrates
(ca. 396–314)
Speusippus
(ca. 410–339)
    Lyceum
Dicaearchus
(ca.370–300)
Aristoxenus
(ca. 370–300)
Eudemus
(ca.370–300)
Theophrastus
(372–288)

Aristotle
(384–322)
400 BCE Plato
(427–347)
500 BCE Pythagoras 
(570–490)

 

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