Jurasovaib's List: From Pythagoras to Einstein

• Pythagorean proof

• he Pythagorean Theorem was known long before Pythagoras, but he may well have been the first to prove it.^[6] In any event, the proof attributed to him is very simple, and is called a proof by rearrangement.

   

  The two large squares shown in the figure each contain four identical triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the white space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean Theorem, Q.E.D.^[7]

   

  That Pythagoras originated this very simple proof is sometimes inferred from the writings of the later Greek philosopher and mathematician Proclus.^[8] Several other proofs of this theorem are described below, but this is known as the Pythagorean one.

• Pythagorean triples

   

  Main article: Pythagorean triple

   

  A Pythagorean triple has three positive integers a, b, and c, such that a² + b² = c². In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths.^[1] Evidence from megalithic monuments in Northern Europe shows that such triples were known before the discovery of writing. Such a triple is commonly written (a, b, c). Some well-known examples are (3, 4, 5) and (5, 12, 13).

   

  A primitive Pythagorean triple is one in which a, b and c are coprime (the greatest common divisor of a, b and c is 1).

   

  The following is a list of primitive Pythagorean triples with values less than 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)

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• The speed of light in vacuum, commonly denoted c, is a universal physical constant important in many areas of physics. Its value is exactly 299,792,458 metres per second, a figure that is exact because the length of the metre is defined from this constant and the international standard for time.^[1]
• The first quantitative estimate of the speed of light was made in 1676 by Rømer (see Rømer's determination of the speed of light).^[83][84] From the observation that the periods of Jupiter's innermost moon Io appeared to be shorter when the Earth was approaching Jupiter than when receding from it, he concluded that light travels at a finite speed, and estimated that it takes light 22 minutes to cross the diameter of Earth's orbit. Christiaan Huygens combined this estimate with an estimate for the diameter of the Earth's orbit to obtain an estimate of speed of light of 220000 km/s, 26% lower than the actual value.^[105]
• Beginning in the 1880s several experiments were performed to try to detect this motion, the most famous of which is the experiment performed by Albert A. Michelson and Edward W. Morley in 1887.^[128] The detected motion was always less than the observational error. Modern experiments indicate that the two-way speed of light is isotropic (the same in every direction) to within 6 nanometres per second.^[129]
• In 1972, using the laser interferometer method and the new definitions, a group at NBS in Boulder, Colorado determined the speed of light in vacuum to be c = 299792456.2±1.1 m/s. This was 100 times less uncertain than the previously accepted value. The remaining uncertainty was mainly related to the definition of the metre.^{[Note 8][104]} As similar experiments found comparable results for c, the 15th Conférence Générale des Poids et Mesures (CGPM) in 1975 recommended using the value 299792458 m/s for the speed of light.^[136]

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• Ole Christensen Rømer (Danish pronunciation: [o(ː)lə ˈʁœːˀmɐ]; 25 September 1644, Århus – 19 September 1710, Copenhagen) was a Danish astronomer who in 1676 made the first quantitative measurements of the speed of light.
• Rømer and the speed of light

• He received the 1921 Nobel Prize in Physics "for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect".^[5]
• He received the 1921 Nobel Prize in Physics "for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect".^[5]

• On July 12, 1939, a few months before the beginning of World War II in Europe, Szilárd and Wigner visited Einstein^[74] and they explained the possibility of atomic bombs, to which pacifist Einstein replied: Daran habe ich gar nicht gedacht ("I had not thought of that at all").^[75] Einstein was persuaded to lend his prestige by writing a letter with Szilárd to President Franklin D. Roosevelt to alert him of the possibility. The letter also recommended that the U.S. government pay attention to and become directly involved in uranium research and associated chain reaction research.

   

  The letter is believed to be "arguably the key stimulus for the U.S. adoption of serious investigations into nuclear weapons on the eve of the U.S. entry into World War II".^[76] In addition to the letter, Einstein used his connections with the Belgian Royal Family^[77] and the Belgian queen mother^[72] to get access with a personal envoy to the White House's Oval Office.^[72] President Roosevelt could not take the risk of allowing Hitler to possess atomic bombs first. As a result of Einstein's letter and his meetings with Roosevelt, the U.S. entered the "race" to develop the bomb, drawing on its "immense material, financial, and scientific resources" to initiate the Manhattan Project. It became the only country to successfully develop an atomic bomb during World War II.

   

  For Einstein, "war was a disease ... [and] he called for resistance to war." By signing the letter to Roosevelt he went against his pacifist principles.^[78] In 1954, a year before his death, Einstein said to his old friend, Linus Pauling, "I made one great mistake in my life—when I signed the letter to President Roosevelt recommending that atom bombs be made; but there was some justification—the danger that the Germans would make them ..."^[79]

   

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• Pythagoras of Samos (/pɪˈθæɡərəs/; Ancient Greek: Πυθαγόρας ὁ Σάμιος Pythagóras ho Sámios “Pythagoras the Samian”, or simply Πυθαγόρας; Πυθαγόρης in Ionian Greek; c. 570 BC – c. 495 BC)^[1][2] was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism.
• 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,
• While the theorem that now bears his name was known and previously utilized by the Babylonians and Indians, he, or his students, are often said to have constructed the first proof. It must, however, be stressed that the way in which the Babylonians handled Pythagorean numbers implies that they knew that the principle was generally applicable, and knew some kind of proof, which has not yet been found in the (still largely unpublished) cuneiform sources.^[47]
• According to legend, the way Pythagoras discovered that musical notes could be translated into mathematical equations was when he passed blacksmiths at work one day and thought that the sounds emanating from their anvils were beautiful and harmonious and decided that whatever scientific law caused this to happen must be mathematical and could be applied to music. He went to the blacksmiths to learn how the sounds were produced by looking at their tools. He discovered that it was because the hammers were "simple ratios of each other, one was half the size of the first, another was 2/3 the size, and so on."
• Another belief attributed to Pythagoras was that of the "harmony of the spheres". Thus the planets and stars moved according to mathematical equations, which corresponded to musical notes and thus produced a symphony.^[51]

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• Order of magnitude[edit]

   

  Rømer starts with an order of magnitude demonstration that the speed of light must be so great that it takes much less than one second to travel a distance equal to Earth's diameter.

   

  The point L on the diagram represents the second quadrature of Jupiter, when the angle between Jupiter and the Sun (as seen from Earth) is 90°.^{[note 6]} Rømer assumes that an observer could see an emergence of Io at the second quadrature (L), and also the emergence which occurs after one orbit of Io around Jupiter (when the Earth is taken to be at point K, the diagram not being to scale), that is 42½ hours later. During those 42½ hours, the Earth has moved further away from Jupiter by the distance LK: this, according to Rømer, is 210 times the Earth's diameter.^{[note 7]} If light travelled at a speed of one Earth-diameter per second, it would take 3½ minutes to travel the distance LK. And if the period of Io's orbit around Jupiter were taken as the time difference between the emergence at L and the emergence at K, the value would be 3½ minutes longer than the true value.

   

  Rømer then applies the same logic to observations around the first quadrature (point G), when Earth is moving towards Jupiter. The time difference between an immersion seen from point F and the next immersion seen from point G should be 3½ minutes shorter than the true orbital period of Io. Hence, there should be a difference of about 7 minutes between the periods of Io measured at the first quadrature and those measured at the second quadrature. In practice, no difference is observed at all, from which Rømer concludes that the speed of light must be very much greater than one Earth-diameter per second.^[5]

• However Rømer also realised that any effect of the finite speed of light would add up over a long series of observations, and it is this cumulative effect that he announced to the Royal Academy of Sciences in Paris. The effect can be illustrated with Rømer's observations from spring 1672.

   

  Jupiter was in opposition on 2 March 1672: the first observations of emergences were on 7 March (at 07:58:25) and 14 March (at 09:52:30). Between the two observations, Io had completed four orbits of Jupiter, giving an orbital period of 42 hours 28 minutes 31¼ seconds.

   

  The last emergence observed in the series was on 29 April (at 10:30:06). By this time, Io had completed thirty orbits around Jupiter since 7 March: the apparent orbital period is 42 hours 29 minutes 3 seconds. The difference seems minute – 32 seconds – but it meant that the emergence on 29 April was occurring a quarter-hour after it would have been predicted. The only alternative explanation was that the observations on 7 and 14 March were wrong by two minutes.