
The carbon age began in 1987 when Jimmy Connors switched from a steel to a carbon racquet.
Discovery Yield Density (year) Strength (g/cm^{3}) (GPa) Gold Ancient .20 19.3 Silver Ancient .10 10.5 Copper 5000 .12 9.0 Bronze 3200 .20 9 Copper + Tin. Stronger than copper Brass 2000 .20 9 Copper + Zinc. Stronger than copper Iron 1200 .25 7.9 In the form of steel. Stronger than bronze and brass Carbon 1963 1.4 1.75 Royal Aircraft develops the first commercial carbon fiber
BCE: 3200 Mesopotamians develop the first known written language 2900 Bronze age begins in Mesopotamia, replaces stone age 2700 1450 Minoan civilization in Crete ~2600 Imhotep, an Egyptian engineer, constructs the first pyramid. 1600 1100 Mycenaean civilization in Crete ~1600 Thera volcano explodes in the Aegean Sea. ~1250 Hercules Trojan war, sometime between 1194 and 1184 BC. 1100 Collapse of the Greek bronze age 1100 800 Greek dark age 776 First Olympic games. Reemergence of Greek civilization. Iron age replaces bronze age 624 546 Thales of Miletus. Philosopher 570 495 Pythagoras of Croton 563 483 Gautama Buddha 536 520 Milo of Croton dominates Olympic wrestling 534 First theatre competition in Athens, won by Thespis 490 Battle of Marathon. Athenians defeat the Persians 480 Thermopylae. 300 Spartans defeat one million Persians 508 Roman Republic founded, replacing monarchy rule 490 First Persian invasion. Battle of Marathon. Athenians defeat the Persians 480 Second Persian invasion. Battle of Thermopylae. 300 Spartans humble the Persian army. 480 Battle of Salamis. Greeks defeat the Persian fleet 479 Battle of Plataea. PanGreek force defeats the Persians. 479 Battle of Mycale. Greeks defeat the remnants of the Persian navy 469 399 Socrates 464 An earthquake strikes Sparta. Subsequent events lead to war with Athens. 460 War commences between Sparta and Athens and continues off and on until 405. 430 Plague of Athens 428 348 Plato 405 Sparta defeats Athens at the battle of Aegospotami. Athens declines hereafter. 395 387 Corinthian war. Hereafter Sparta dominates Greece until the battle of Leuctra 384 322 Aristotle 371 Battle of Leuctra. Thebes and allies defeat Sparta. 362 Battle of Mantinea. Thebes & allies defeat Sparta. Thebes and Sparta decline hereafter and Macedonia rises in power. 356 323 Alexander 331 Battle of Gaugamela. Alexander's Macedonian and Greek army defeats the Persians 350 280 Demetrius. Student of Aristotle. Founded Library of Alexandria 310 230 Aristarchus. Measured the size and distance of the moon and the sun. 287 212 Archimedes 275 195 Eratosthenes. Chief of the Alexandrian Library. 202 Battle of Zama. Rome defeats Carthage. 197 Rome defeats Macedon at the Battle of Cynoscephalae. Rome dominates Greece hereafter. 190 120 Hipparchus, astronomer 138 78 Sulla 100 44 Julius Caesar 73 Revolt of Roman slaves led by Spartacus and Crixus 70 19 Virgil 63 14 Octavian 60 53 First Triumvirate (Caesar, Pompey, Crassus) 53 Battle of Carrhae. Crassus defeated by the Persians. 48 Battle of Pharsalus. Julius Caesar and Mark Antony defeat Pompei. 44 Assassination of Julius Caesar 43 33 Second Triumvirate (Octavian, Antony, Lepidus). End of Roman Republic 42 Battle of Philippi. Octavian and Antony defeat Brutus and Cassius. 31 Octavian becomes emperor 30 Octavian defeats Antony and Cleopatra at Alexandria 27 Pax Romana begins and lasts until 180 CE CE: 90 168 Ptolemy. Library of Alexandria. 476 550 Aryabhata, Indian mathematician. Refines Aristarchus' measurements of the size of the sun. 780 850 Muhammad al Khwarizmi. Mathematician in Baghdad. 872 950 AlFarabi, mathematician 1206 Ghenghis Khan becomes leader of the Mongolians. Mongolian empire begins 1258 Baghdad, a city of scholars, is destroyed by Mongolian invaders. 1473 1543 Nicolaus Copernicus 1543 Copernicus publishes the heliocentric model of the solar system. 1546 1601 Tycho Brahe does precision measurements of the orbits of the planets. 1561 1626 Francis Bacon. Introduces empiricism in science. Influential in the founding of the Royal Society 1571 1630 Johannes Kepler. Discovers that planets orbit in ellipses. 1564 1642 Galileo. Develops the concept of acceleration. 1632 1723 Christopher Wren. Builder of cathedrals and founder of the Royal Society 1643 1727 Isaac Newton. Develops laws of inertia, gravity, and The Calculus. 1660 Royal Society founded 1666 French Academy of Sciences founded 1707 1783 Leonhard Euler 1777 1855 Carl Friedrich Gauss 1800 Volta invents the battery 1831 1879 James Clerk Maxwell. Unifies the theories of electricity and magnetism 1864 Maxwell develops electromagnetism 1905 Einstein develops the theory of special relativity for relativistic motion 1909 Rutherford discovers the nucleus 1915 Einstein develops the theory of general relativity for gravitation 1926 Schroedinger discovers the "Schroedinger formula" for quantum mechanics 1927 Quantum mechanics developed. 1947 Transistor invented. Silicon age begins 1987 Jimmy Connors switches from a steel to a graphite racket. Carbon age begins
BCE: 5000 Copper age begins 3200 Bronze age begins. Bronze = Copper + Tin 3200 Mesopotamians develop the first known written language 2900 Bronze age begins in Mesopotamia, replaces stone age ~2600 Imhotep, an Egyptian engineer, constructs the first pyramid 1200 Iron age begins 776 First Olympic games. Reemergence of Greek civilization 624 546 Thales of Miletus. Philosopher 570 495 Pythagoras of Croton 469 399 Socrates 428 348 Plato 384 322 Aristotle 350 280 Demetrius. Student of Aristotle. Founded Library of Alexandria 310 230 Aristarchus. Measured the size and distance of the moon and the sun. 287 212 Archimedes 275 195 Eratosthenes. Chief of the Alexandrian Library. 190 120 Hipparchus, astronomer CE: 90 168 Ptolemy. Library of Alexandria. 476 550 Aryabhata, Indian mathematician. Refines Aristarchus' measurements of the size of the sun. 780 850 Muhammad al Khwarizmi. Mathematician in Baghdad. 872 950 AlFarabi, mathematician 1286 Eyeglasses invented in Italy 1543 Copernicus publishes the heliocentric model of the solar system. 1561 1626 Francis Bacon. Introduces empiricism in science. Influential in the founding of the Royal Society 1564 1642 Galileo. Develops the concept of acceleration. 1601 Tycho Brahe completes observations of the positions of the planets 1609 Kepler finds that planets orbit as ellipses, using Brahe's data 1632 1723 Christopher Wren. Builder of cathedrals and founder of the Royal Society 1643 1727 Isaac Newton. Develops laws of inertia, gravity, and The Calculus. 1660 Royal Society founded 1666 French Academy of Sciences founded 1707 1783 Leonhard Euler 1777 1855 Carl Friedrich Gauss 1800 Volta invents the battery 1864 Maxwell develops the theory of electromagnetism 1900 Planck publishes the quantum hypothesis 1905 Einstein develops the theory of special relativity for relativistic motion 1909 Rutherford discovers the nucleus 1915 Einstein develops the theory of general relativity for gravitation 1926 Schroedinger discovers the "Schroedinger equation" for quantum mechanics 1927 Quantum mechanics developed. 1947 Transistor invented. Silicon age begins 1987 Jimmy Connors switches from a steel to a graphite racket. Carbon age begins 2010 Smart phone age begins
510 Cleisthenes establishes democracy in Athens upon the overthrow of King Hippias 508 Roman Republic founded, replacing monarchy rule 27 Pax Romana begins and lasts until 180 CE 1215 English Magna Carta 1660 Royal Society founded by Christopher Wren 1666 French Academy of Sciences founded 1688 England, Revolutionary Settlement. Parliament enacts laws limiting the power of the king. 1689 England, Bill of Rights and Mutiny Bill 1694 England, Triennial Bill 1696 England, Treason Act 1701 England, Act of Settlement 1769 Golden age of the American Philosophical Society begins as Benjamin Franklin becomes president. Members included Jefferson, Washington, Hamilton, Adams, Madison, Paine, and McHenry. 1788 American Constitution established 1791 American Bill of Rights 1810 Modern era of British Prime Ministers begins when King George III goes insane 1830 England, Westminster system of government established (Parliament) 1911 England, Parliament Act, lessening the power of the House of Lords 1949 British Parliaments acts of 1949 2014 British House of Lords Reform Act of 2014
1312 Europeans arrive at The Canary Islands Unknown 1351 Azores. First appearance on a map. Unknown 1427 Azores Islands, Portugal de Silves 1419 Madeira Islands, Portugal Zarco & Teixeira 1456 Cape Verde de Noli 1487 Southern tip of Africa Dias 1492 West Indies Columbus 1494 Treaty of Tordesillas 1497 Newfundland Caboto 1498 India Da Gama 1500 Brazil Cabral 1501 Greenland (first modern mapping) CorteReal 1501 Newfoundland CorteReal 1505 Bermuda de Bermudez 1512 Spice Islands de Abreu 1520 Strait of Magellan Magellan 1520 Conquest of the Aztec Empire Cortez 1522 Westward circumnavigation of the world Magellan 1526 Marshall Islands de Salazar 1533 Conquest of the Incan Empire Pizarro 1565 Eastward crossing of the Pacific de Legazpi Passed north of Hawaii 1574 Archipellago Juan Fernandez Islands 1574 Desventuradas Islands 1606 Australia Janszoon 1606 Pitcairn Islands 1642 New Zealand and Tasmania Tasman 1722 Easter Island Roggeveen 1750 Cumberland gap Walker 1793 Isla Salas y Gomez 1778 Hawaii Cook 1820 Antarctica Bransfield 1859 Midway Middlebrooks
~808 Qing Xuzi publishes a formula resembling gunpower, consisting of 6 parts sulfur, 6 parts saltpeter, and 1 part birthwort herb (for carbon). ~850 Incendiary property of gunpower discovered 1132 "Fire lances" used in the siege of De'an, China 1241 Mongols use firearms at the Battle of Mohi, Hungary 1338 Battle of Arnemuiden. First naval battle involving cannons. The English had 3. 1346 Cannons used in the Siege of Calais and the Battle of Crecy 1540 Biringuccio publishes "De la pirotechnia", giving recipes for gunpowder 1610 First flintlock rifle 1774 Lavoisier appointed to develop the French gunpowder program. By 1788 French gunpowder was the best in the world. 1884 Vieille invents smokeless gunpowder, which was 3 times more powerful than black powder and less of a nuisance on the battlefield.
2650 2600 Imhotep 570 495 Pythagoras 310 230 Aristarchus 276 194 Eratosthenes 287 212 Archimedes 90 168 Ptolemy 476 550 Aryabhata 1452 1519 da Vinci 1473 1543 Copernicus 1501 1576 Cardano 1546 1601 Brahe 1548 1600 Bruno 1548 1620 Stevin 1550 1617 Napier 1561 1626 Bacon 1564 1642 Galileo 1571 1630 Kepler 1580 1626 Snellius 1596 1650 Descartes 1601 1665 Fermat 1623 1662 Pascal 1627 1691 Boyle 1629 1695 Huygens 1630 1677 Barrow 1632 1723 Wren 1635 1703 Hooke 1643 1727 Newton 1646 1716 Leibniz 1656 1742 Halley 1700 1782 Bernoulli, Daniel 1707 1783 Euler 1731 1810 Cavendish 1777 1855 Gauss 1791 1867 Faraday 1811 1832 Galois 1821 1894 Helmholtz 1831 1879 Maxwell 1844 1906 Boltzmann 1845 1918 Cantor 1850 1925 Heaviside 1853 1925 RicciCurbastro 1853 1928 Lorentz 1854 1912 Poincare 1862 1943 Hilbert 1858 1947 Planck 1871 1937 Rutherford 1879 1955 Einstein 1885 1962 Bohr 1887 1961 Schroedinger 1898 1964 Szilard 1900 1958 Pauli 1901 1954 Fermi 1901 1976 Heisenberg 1902 1984 Dirac 1903 1957 von Neumann 1906 1978 Godel 1906 1998 Weil 1908 1968 Landau 1913 1996 Erdos 1918 1988 Feynman 1928 2014 Grothendieck 1951 Witten
The ancient metals such as iron, copper, tin, and zinc are obtained by carbon smelting minerals. Cobalt was the first metal discovered since iron and it's discovery inspired people to smelt every known mineral in the hope of yielding a new metal. By 1800 nearly all of the carbon smeltable metals had been discovered.
Some elements can't be carbon smelted and require electrolysis to isolate. Electrochemistry began in 1800 with the invention of the battery and most of the remaining metals were discovered soon after. Sodium and potassium were isolated by electrolysis in 1807 and these were used to smelt metals that couldn't be smelted with carbon.
Discovery Method of Abundance in (year) discovery crust (ppm) Carbon Ancient Naturally occuring 400 Coal, diamond Gold Ancient Naturally occuring .0031 Silver Ancient Naturally occuring .08 Sulfur Ancient Naturally occuring 420 Lead 6500 Smelt with carbon 10 Copper 5000 Smelt with carbon 68 Bronze (As) 4200 Copper + Arsenic Bronze (Sn) 3200 Copper + Tin Tin 3200 Smelt with carbon 2.2 Brass 2000 Copper + Zinc Mercury 2000 Heat the oxide .067 Iron 1200 Smelt with carbon 63000 In the form of steel Zinc 1300 Smelt with carbon 79 Date when first produced in pure form Antimony 1540 Smelt with iron .2 Arsenic 1649 Heat the oxide 2.1 Phosphorus 1669 Heat the oxide 10000 Cobalt 1735 Smelt with carbon 30 First metal discovered since iron Platinum 1735 Naturally occuring .0037 Nickel 1751 Smelt with carbon 90 Hydrogen 1766 Hot iron + steam 1500 Oxygen 1771 Heat HgO 460000 Nitrogen 1772 From air 20 Manganese 1774 Smelt with carbon 1120 Molybdenum 1781 Smelt with carbon 1.1 Tungsten 1783 Smelt with carbon 1100 Chromium 1797 Smelt with carbon 140 Palladium 1802 Chemistry .0063 Osmium 1803 .0018 Iridium 1803 .004 Rhodium 1804 Smelt with zinc .0007 Smelt Na_{3}RhCl_{6} with zinc Sodium 1807 Electrolysis 23000 Potassium 1807 Electrolysis 15000 Magnesium 1808 Electrolysis 29000 Cadmium 1817 Smelt with carbon .15 Lithium 1821 Electrolysis 17 Zirconium 1824 Smelt with potassium Aluminum 1827 Smelt with potassium 82000 Silicon 1823 Smelt with potassium 270000 Beryllium 1828 Smelt with potassium 1.9 Thorium 1929 Smelt with potassium Vanadium 1831 190 Uranium 1841 Smelt with potassium 1.8 Ruthenium 1844 Smelt with carbon .001 Tantalum 1864 Smelt with hydrogen 1.7 Niobium 1864 Smelt with hydrogen 17 Fluorine 1886 Electrolysis 540 Helium 1895 From uranium ore Titanium 1910 Smelt with sodium 66000 Hafnium 1924 Rhenium 1928 From molybdenite .0026 Scandium 1937 26Gold was the densest element known until the discovery of platinun in 1735. It was useful as an uncounterfeitable currency until the discovery of tungsten in 1783, which has the same density as gold.
Wood fires are 200 Celsius short of the copper smelting temperature. Coal has to be used.
Titanium can't be smelted with carbon because it produces titanium carbide (TiC).
The usefulness of a metal as a sword depends on its strength/density ratio. The table below shows all the metals with a ratio of at least 5 MJoules/kg. For these metals, strength tends to be proportional to density and the strength/density ratio has a characteristic value of 10 MJoules/kg. Beryllium is the sole outlier with a superlatively large value of 71 MJoules/kg.
The low density metals are ones up to vanadium on the periodic table. None are carbon smeltable and all require electrochemistry to isolate. The first low density metal to be produced was magnesium in 1808.
Protons Strength Density Strength/Density Carbon Discovery (GPa) (g/cm^{3}) (MJoule/kg) smeltable year Beryllium 4 132 1.85 71.4 no 1828 Magnesium 12 17 1.74 9.8 no 1808 Aluminum 13 26 2.70 9.6 no 1827 Scandium 21 29 3.0 9.7 no 1937 Titanium 22 44 4.5 9.8 no 1910 Vanadium 23 47 6.0 7.8 no 1831 Chromium 24 115 7.2 16.0 yes 1797 Manganese 25 75 7.2 10.4 yes 1774 Iron 26 82 7.9 10.4 yes 1200 Cobalt 27 75 8.9 8.4 yes 1735 Nickel 28 76 8.9 8.5 yes 1751 Copper 29 48 9.0 5.3 yes 5000 Zinc 30 43 7.1 6.0 yes 1746 Molybdenum 42 120 10.3 11.7 yes 1781 Ruthenium 44 173 12.4 14.0 yes 1844 Rhodium 45 150 12.4 12.1 yes 1804 Tungsten 74 161 19.2 8.4 yes 1783 Rhenium 75 178 21.0 8.5 yes 1928 Osmium 76 222 22.6 9.8 yes 1803 Iridium 77 210 22.6 9.3 yes 1803 Uranium 92 111 19.1 5.8 no 1841 Strength: Shear modulus (GPascals) Density: Density (grams/cm^{3}) Strength/Density: Shear modulus / Density (MJoules/kg)
For a metal, the stiffness is characterized by the "shear strength" and the sword worthiness is characterized by the shear strength over the density (the "strength to weight ratio"). For example for iron,
Shear modulus = S = 82 GJoules/meter^{3} Density = D = 7900 kg/meter^{3} Sword worthiness = Q = S/D = 10.4 MJoules/kg
This plot includes all metals with a strength/density at least as large as lead, plus mercury. Beryllium is beyond the top of the plot.
600 Wootz steel developed in India and is renowned as the finest steel in the world. 1700 The technique for making Wootz steel is lost. 1790 Wootz steel begins to be studied by the British Royal Society. 1838 Anosov replicates Wootz steel.Wootz steel is a mix of two phases: martensite (crystalline iron with .5% carbon), and cementite (iron carbide, Fe_{}, 6.7% carbon).
In prehistoric times iron meteorites were the only source of metallic iron. They consist of 90% iron and 10% nickel.
240 Eratosthenes measures the Earth's circumference to 20% error. 240 Aristarchus proves that the sun is at least 10 times larger than the Earth using lunar eclipses. 150 Ptolemy publishes The Almagest with the geocentric model 550 Aryabhata publishes accurate measurements of size of the sun and moon 1543 Copernicus publishes a heliocentric model. 1600 Brahe measures accurate planet positions 1608 Lippershey invents the telescope 1609 Galileo builds a telescope and begins observing 1609 Kepler proves that planets orbit as ellipses using Brahe's data 1613 Galileo publishes observations of the phases of Venus, which support the heliocentric model 1632 Galileo publishes the "Dialogue Concerning the Two Chief World Systems", which contained a comparison of the systems of Ptolemy and Copernicus 1672 Richter and Cassini measure the parallax of Mars, producing a precise value for the size of the sun 1687 Newton publishes the Principia Mathematica, which contained the calculus, the laws of motion (F=MA), and a proof that planets orbit as ellipses. 1718 Halley finds that the stars move. He found that Sirius, Arcturus and Aldebaran were 1/2 of a degree from the positions charted by the Ancient Greek astronomer Hipparchus 1783 Herschel finds that the solar system is moving with respect to the stars 1826 Olbers' paradox. If the stars in the universe are uniformly distributed and if the universe is infinite, then the sky would appear infinitely bright with stars. 1863 Bessel measures the first stellar parallax, showing that the stars are more than 4 light years away. This also implies that stars are as luminous as the sun. The parallax of stars is too small to see without a telescope. 1905 Einstein publishes special relativity 1915 Einstein publishes the general theory of relativity. Einstein shows that general relativity is consistent with the existence of a cosmological constant. At the time the cosmological constant was a proposed explanation for why the universe hasn't collapsed gravitationally. 1920 Shapley finds that the sun is not at the center of the galaxy. Because starlight is absorbed by interstellar gas, we only see the nearby stars and it appears as though we live at the center of a disk of stars. Shapley measured the distances to globular clusters and found that they are centered on a point (the galactic center) that is far from the sun. 1922 Friedmann finds a solution to the equations of general relativity that are consistent with an expanding universe. 1923 Hubble measures the distances to Andromeda and Triangulum and finds that they are outside the Milky Way. These were the first objects to be shown to be outside the Milky Way. 1929 Hubble's law published. For distant galaxies, the recession velocity is proportional to distance. 1933 Zwicky's analysis of the Coma cluster of galaxies shows that they contain unseen matter that is not due to stars. 1965 Penzias and Wilson discover the cosmic microwave background radiation. 1970 Rubin and Ford measure galactic rotation and show that galaxies contain matter that is not due to stars. 1980 Guth and Starobinsky propose the theory of inflation to explain why the universe is flat 1998 Observations of supernovae show that the expansion of the universe is accelerating and the the cosmological constant is positive. Previous to this it was not known if the universe was destined for collapse (big crunch) or for infinite expansion (big chill). 2003 WMAP mission measures the Hubble constant to 5% precision, as well as other cosmological parameters. Previous to this, the Hubble constant had an error of ~ 20%. This settled once and for all the question of the overall structure of the universe.
260 Aristarchus established that the distance to the sun is at least 20 times the distance to the moon.
In 499, Aryabhata publishes a measurement of the distance to the sun.
Brahe's data consisted of measurements of angles between different objects. This data could be used to establish the shape of orbits but not their size. For example, if the size of the solar system were doubled along with the speeds of the planets, the angles would stay the same and you wouldn't be able to tell the difference.
In 1639, Horrocks used a transit of Venus to measure the distance to the sun, but this method is incapable of giving an accurate value, and it can only be done once per century.
In 1672, Richter and Cassini measured the parallax of Mars which gives a result for the distance to the sun that is more accurate than the Venus method. The Mars method has an advantage over the Venus method in that it can be done once every 26 months, when Mars is at closest approach.
In 1676, Romer used the moons of Jupiter to measure the time it takes for light to cross the Earth's orbit. This gives a value for R/C.
In 1729, Bradley measured the deflection of starlight due to the Earth's motion, which gives a measurement of V/C, or equivalently, a measurement of R/C.
In 1849, Fizeau produced the first measurement for the speed of light that was independent of the Earthsun distance R.
Speed of light = C Earthsun distance = R Earth orbital velocity = V Earth orbital time (1 year) = T = 2 π R / V Time for light to cross the Earth's orbit= t = 2 R / C
1585 Simon Stevin introduces decimal numbers to Europe. (for example, writing 1/4 as .25) 1586 Simon Stevin drops objects of varying mass from a church tower to demonstrate that acceleration is independent of mass. 1604 Galileo publishes the mathematical description of acceleration. 1614 Logarithms invented by John Napier, making possible precise calculations of equal tuning ratios. Stevin's calculations were mathematically sound but the frequencies couldn't be calculated with precision until logarithms were developed. 1637 Cartesian geometry published by Fermat and Descartes. This was the crucial development that triggered an explosion of mathematics and opened the way for the calculus. 1684 Leibniz publishes the calculus 1687 Newton publishes the Principia Mathematica, which contained the calculus, the laws of motion (F=MA), and a proof that planets orbit as ellipses. 1733 Euler develops the calculus of variations 1762 Lagrange discovers the divergence theorem, the 2D generalization of the fundamental theorem of calculus. The surface flux integral equals the volume divergence integral. 1788 Lagrangian mechanics published 1821 Cauchy publishes the "epsilondelta" definition of a limit, raising the level of rigor in mathematics. 1822 Fourier transform published 1828 Green's theorem. In 2D, the circulation line integral equals tthe curl area integral 1833 Hamiltonian mechanics published 1834 Eikonal approximation developed by Hamilton 1850 KelvinStokes theorem. 3D generalization of Green's theorem 1854 Stokes theorem. Generalization of the KelvinStokes theorem 1854 Riemann Integral published, the first rigorous definition of an integral 1854 Chebyshev polynomials published 1863 Helmholtz publishes "On the Sensations of Tone" 1870 Heine defines "uniform continuity" 1872 Heine proves that a continuous function on an open interval need not be uniformly continuous. 1872 Weierstrass publishes the "Weierstrass function", the first example of a function that is continuous everywhere and differentiable nowhere. 1877 Lord Rayleigh publishes "Theory of Sound" 1887 Poincare discovers the phenomenon of chaos while studying celestial mechanics 1926 WKB theory published 1978 "Bender & Orszag" textbook published. Art of blending special functions like Scotch.
2000 System of hours, minutes, and seconds developed in Sumer 300 Water clock developed in Ancient Greece 100 Zhang Heng constructs a seismometer using pendulums that was capable of detecting the direction of the Earthquake. 1300 First mechanical clock developed. 1400 Springbased clocks developed. 1500 Pendulums are used for power, for machines such as saws, bellows, and pumps. 1582 Galileo finds that the period of a pendulum is independent of mass and oscillation angle, if the angle is small. 1636 Mersenne and Descartes find that the pendulum was not quite isochronous. Its period increased somewhat with its amplitude. 1656 Huygens builds the first pendulum clock, delivering a precision of 15 seconds per day. Previous devices had a precision of 15 minutes per day. Fron this point on pendulum clocks were the most accurate timekeeping devices until the development of the quartz oscillator was developed in 1921. 1657 Balance spring developed by Hooke and Huygens, making possible portable pocketwatches. 1658 Huygens publishes the result that pendulum rods expand when heated. This was the principal error in pendulum clocks. 1670 Previous to 1670 the verge escapement was used, which requires a large angle. The anchor escapement mechanism is developed in 1670, which allows for a smaller angle. This increased the precision because the oscillation period is independent of angle for small angles. 1673 Huygens publishes a treatise on pendulums. 1714 The British Parliament establishes the "Longitude Prize" for anyne who could find an accurate method to determine longitude at sea. At the time there was no clock that could measure time on a moving ship accurately enough to determine longitude. 1721 Methods are developed for compensating for thermal expansion error of a pendulum. 1726 Gridiron pendulum developed, improving precision to 1 second per day. 1772 Harrison builds a clock which James Cook used in his exploration of the Pacific. Cook's log is full of praise for the watch and the charts of the Pacific Ocean were remarkably accurate. 1772 Harrison gives one of his clocks to King George III, who personally tested it and found it to be accurate to 1/3 of one second per day. King George III advised Harrison to petition Parliament for the full Longitude Prize after threatening to appear in person to dress them down. 1851 Foucault shows that a pendulum can be used to measure the rotation period of the Earth. The penulum swings in a fixed frame and the Earth rotates with respect to this frame. In the Earth frame the pendulum appears to precess. 1921 Quartz electronic oscillator developed 1927 First quartz clocks developed, which were more precise than pendulum clocks. Length of the pendulum = L Gravity constant = g = 9.8 meters/second^{2} Angle of oscillation = Z (radians) Period of the pendulum = T = 2 π (L/g)^{½} For small angles (Z << 1)As the angle increases the period of oscillation increases.
~808 Qing Xuzi publishes a formula resembling gunpower, consisting of 6 parts sulfur, 6 parts saltpeter, and 1 part birthwort herb (for carbon). ~850 Incendiary property of gunpower discovered 1540 Biringuccio publishes "De la pirotechnia", giving recipes for gunpowder 1661 Boyle publishes "The Sceptical Chymist", a treatise on the distinction between chemistry and alchemy. It contains some of the earliest modern ideas of atoms, molecules, and chemical reaction, and marks the beginning of the history of modern chemistry. 1662 Boyle discovers that for air at fixed temperature, Pressure * Volume = Constant 1663 Guericke invents the first electrostatic generator, which uses mechanical work to separate charge. Generators were refined until they were superceded by the battery. 1671 Boyle discovers that combining iron filings and acid produces hydrogen gas. 1754 Black isolates CO2 1758 Black formulates the concept of latent heat to explain phase transitions 1766 Cavendish identifies hydrogen as a colorless, odourless gas that burns in air. 1772 Scheele produces pure oxygen gas by heating HgO. 1774 Priestly produces pure oxygen gas by focusing sunlight on HgO. He noted that it is combustible and that it gives energy when breathed. 1745 von Kleist invents the capacitor, a device for storing charge generated by an electrostatic generator. 1746 van Musschenbroek refines the capacitor, which comes to be known as a "Leyden jar". 1777 Lavoisier finds that in the reaction tin+oxygen, mass is conserved. He also finds that oxygen is not the only component of air, that air also consists of something else. 1780 Galvani observes that when a frog leg is touched by an iron scalpel, it twitches. This was the inspiration for Volta to invent the battery. 1781 Cavendish finds that buring hydrogen + oxygen produces water. 1787 Charles finds that for air at constant pressure, Volume = Constant * Temperature He also finds that this applies for O2, N2, H2, and CO2. 1789 Lavoisier publishes "Traite Elementaire de Chimie", the first modern chemistry textbook. It is a complete survey of (at that time) modern chemistry, the law of conservation of mass. 1791 Volta develops the first electrochemical cell, consisting of two different metals separated by a moist intermediary. 1797 Proust proposes the law of definite proportions, that elements combine in small whole number ratios to form compounds. 1800 Volta constructs the first "battery" by connecting multiple electrochemical cells in parallel, increasing the output power and voltage. 1800 Volta constructs the first battery, a set of electrochemical cells wired in serial to increase the voltage. 1801 Dalton publishes the law of partial pressures. The pressure of a mix of gases is equal to the sum of the pressures of the components. He also finds that when a light and heavy gas are mixed, the heavy gas does not drift to the bottom but rather fills the space uniformly. 1805 GayLussac and Humboldt find that water is formed of two volumes of hydrogen gas and one volume of oxygen gas. 1809 GayLussac finds that for an ideal gas at constant volume, Pressure = Constant * Temperature 1811 Avogadro finds that equal volumes of different gases have the same number of particles. At constant temperature and pressure, Volume = Constant * NumberOfParticles 1811 Avogadro arrives at the correct interpretation of water's composition, based on what is now called Avogadro's law and the assumption of diatomic elemental molecules 1840 Hess finds that energy is conserved in chemical reactions 1848 Lord Kelvin establishes concept of absolute zero, the temperature at which all molecular motion ceases. 1860 Cannizzaro publishes a table of atomic weights of the known elements 1864 Gulberg and Waage propose the law of mass action 1869 Mendeleev publishes a periodic table containing the 66 known elements 1876 Gibbs publishes the concept of "Gibbs free energy" 1877 Boltzmann defines entropy and develops thermodynamics 1877 Pictet freezes CO2 and liquefies oxygen. Liquification enables the purification of gases. 1894 Ramsay discovers the noble gases, filling a large and unexpected gap in the periodic table
1635 Gassendi measures the speed of sound to be 478 m/s with 25% error. 1660 Viviani and Borelli produce the first accurate measurement of the speed of sound, giving a value of 350 m/s. 1660 Hooke's law published. The force on a spring is proportional to the change in length. 1662 Boyle discovers that for air at fixed temperature, Pressure * Volume = Constant Hence, air obey's Hooke's law 1687 Newton publishes the Principia Mathematica, which contains the first analytic calculation of the speed of sound. The calculated value was 290 m/s and the true value is 342 m/s (at 20 Celsius).Newton's result was the first solid evidence for the existence of atoms. His result differed from the correct value because it had not yet been discovered that air heats when compressed. If you add this effect you get the right value.
The reason air heats when compressed is because it is composed of atoms. You can see this in action with the "Gas" simulation at phet.colorado.edu. You can also see how atoms in a gas can carry a sound wave, and why the sound speed has the same orderofmagnitude as the thermal velocity of the atoms.
1789 Lavoisier publishes "Traite Elementaire de Chimie", the first modern chemistry textbook. It is a complete survey of (at that time) modern chemistry, the law of conservation of mass. 1797 Proust proposes the law of definite proportions, that elements combine in small whole number ratios to form compounds. 1801 Dalton publishes the law of partial pressures. The pressure of a mix of gases is equal to the sum of the pressures of the components. He also finds that when a light and heavy gas are mixed, the heavy gas does not drift to the bottom but rather fills the space uniformly. 1805 GayLussac and Humboldt find that water is formed of two volumes of hydrogen gas and one volume of oxygen gas. 1809 GayLussac finds that for an ideal gas at constant volume, Pressure = Constant * Temperature 1811 Avogadro finds that equal volumes of different gases have the same number of particles. At constant temperature and pressure, Volume = Constant * NumberOfParticles 1811 Avogadro arrives at the correct interpretation of water's composition, based on what is now called Avogadro's law and the assumption of diatomic elemental molecules 1860 Cannizzaro publishes a table of atomic weights of the known elements 1869 Mendeleev publishes a periodic table containing the 66 known elements 1877 Boltzmann defines entropy and develops thermodynamicsAll of these results support the hypothesis that matter is composed of atoms, but there was no known experiment sensitive enough to measure the size and mass of an individual atom.
Wavelength of violet light = 4e7 meters Diameter of an iron atom = 2e10 metersViolet photons are much larger than atoms and so you can't see atoms in an optical microscope.
1905 Einstein publishes a method for measuring the mass of an atom using Brownian motion 1908 Perrin uses Einstein's method to produce the first measurement of the mass of an atom. This is equivalent to measuring the value of Avogadro's number.Minutephysics atoms
600 Thales discovers static electricity 600 Thales discovers natural magnets, which are magnetized magnetite (Fe_{3}O_{4}) 250 Magnetic compass invented 1600 Gilbert publishes a treatise on electromagnetism 1663 Guericke invents the first electrostatic generator, which uses mechanical work to separate charge. Generators were refined until they were superceded by the battery. 1745 von Kleist invents the capacitor, a device for storing charge generated by an electrostatic generator. 1746 van Musschenbroek refines the capacitor, which comes to be known as a "Leyden jar". 1748 Franklin determines that there are two types of charge, positive and negative, and that charge is conserved. 1752 Franklin discovers the link between lightning and electricity by using a kite to transfer charge from lightning to a Leyden jar. 1780 Galvani observes that when a frog leg is touched by an iron scalpel, it twitches. This was the inspiration for Volta to invent the battery. 1785 Coulomb discovers the electric force law 1791 Volta develops the first electrochemical cell, consisting of two different metals separated by a moist intermediary. 1800 Volta constructs the first "battery" by connecting multiple electrochemical cells in parallel, increasing the output power and voltage. In 1836 Daniell refined the battery, making it suitable for industrial use 1820 Oersted finds that an electric current produces a magnetic field 1826 Ampere finds that electric currents attract each other 1831 Faraday finds that a changing magnetic field produces an electric field 1861 Maxwell finds that a changing electric field produces a magnetic field 1862 Maxwell unifies the previous discoveries about currents and magnetic fields with "Maxwell's equations" 1864 Maxwell finds that light is an electromagnetic wave This theory predicts a paradox, that the speed of light is invariant 1883 Wimshurst machine invented for using induction to separate charge 1884 Heaviside invents the vector calculus and uses it to simplify Maxwell's equations 1887 Hertz achieves the first detection of electromagnetic waves 1887 MichelsonMorley experiment finds that the speed of light is invariant 1889 Heaviside publishes the force law for a charge moving in a magnetic field 1892 Lorentz discovers the "Lorentz transform" for special relativity This offered an explanation for the MichelsonMorley experiment 1904 Lorentz finds that the Lorentz transform resolves the paradoxes of Maxwell's equations 1905 Einstein and Poincare each publish a complete formulation of the theory of special relativity
1650 Guericke builds the first vacuum pump 1660 Boyle's law for a gas at constant temperature: Pressure * Volume = Constant 1676 Leibniz defines kinetic energy and notes that it is conserved in many mechanical processes 1698 Savery develops a steam engine 1702 Amontons introduces the concept of absolute zero, based on observations of gases 1761 Black discovers that ice absorbs heat without changing its temperature when melting 1776 Smeaton publishes a paper on experiments related to power, work, momentum, and kinetic energy, supporting the conservation of energy 1777 Scheele distinguishes heat transfer by thermal radiation from that by convection and conduction 1791 Prevost shows that all bodies radiate heat, no matter how hot or cold they are 1798 Thompson performs measurements of the frictional heat generated in boring cannons and develops the idea that heat is a form of kinetic energy 1802 GayLussac publishes Charles's law. For a gas at constant pressure, Temperature * Volume = Constant 1802 GayLussac publishes GayLussac's law. For a gas at constant volume, Temperature * Pressure = Constant 1804 Leslie observes that dark surfaces radiate heat more effectively than lightcolored surfaces. 1810 Leslie freezes water to ice artificially 1819 Dulong and Petit find that the heat capacity of a crystal is proportional to the number of atoms 1824 Carnot analyzes the efficiency of steam engines; he develops the notion of a reversible process and, in postulating that no such thing exists in nature, lays the foundation for the second law of thermodynamics, and initiating the science of thermodynamics 1827 Brown discovers the Brownian motion of pollen and dye particles in water 1831 Melloni demonstrates that infrared radiation can be reflected, refracted, and polarized in the same way as light 1834 Clapeyron combines Boyle's Law, Charles's Law, and GayLussac's Law to produce a Combined Gas Law. Pressure * Volume = Constant * Temperature 1842 Mayer calculates the equivalence between heat and kinetic energy 1847 Helmholtz publishes "On the Conservation of Force", where energy is used to connect mechanics, heat, light, electricity and magnetism.
1752 Dalibard uses a lightening rod to validate Franklin's hypothesis. 1909 Millikan and Fletcher use the "Oil drop" experiment to obtain the first measurement of the electron mass and charge. They find that the electron mass is 2000 times smaller than a hydrogen atom. 1947 Franklin finds that electricity is composed of positive and negative charge and that charge is conserved. 1750 Franklin proposes an experiment to prove that lightening is electricity by flying a kite in a lightening storm.
1909 Rutherford, Geiger, and Marsden use the "Rutherford scattering" experiment to show that the nucleus is much smaller than the atom. Size of atom ~ 2e10 meters Size of nucleus ~ 2e15 metersPhet simulation on Rutherford scattering
1635 Gassendi measures the speed of sound to be 478 m/s with 25% error. 1660 Viviani and Borelli produce the first accurate measurement of the speed of sound, giving a value of 350 m/s. 1660 Hooke's law published. The force on a spring is proportional to the change in length. 1662 Boyle discovers that for air at fixed temperature, Pressure * Volume = Constant 1687 Newton publishes the Principia Mathematica, which contains the first analytic calculation of the speed of sound. The calculated value was 290 m/s. The fact that Newton's calculation differed from the measured speed was the first solid clue for the existence of atoms, and it also contained a clue for quantum mechanics. In Newton's age it was not known that a gas heats if compressed. If you include this effect you get the correct value for the speed of sound. The fact that a gas heats when compressed is due to the fact that a gas is composed of atoms. 1859 Kirchhoff finds that the blackbody spectrum depends only on temperature 1877 Boltzmann suggests that the energy levels of a physical system could be discrete based on statistical mechanics and mathematical arguments 1887 Hertz discovers the photoelectric effect, that light can eject electrons from a material 1888 Rydberg measures the emission frequencies of the hydrogen atom 1900 Planck finds if you assume photon energy is quantized, the correct blackbody spectrum emerges 1905 Einstein interprets the photoelectric effect as being caused by discrete packets of light (photons) 1907 Rutherford discovers the nucleus with the "Rutherford scattering" experiment 1909 Taylor demonstrates that the diffraction pattern of light through a double slit is preserved even if the photons are emitted one at a time 1909 Einstein shows that the Planck law implies that photons carry momentum
1803 Young discovers the diffraction of light, suggesting that light is a wave 1861 Maxwell develops the "Maxwell's equations", unifying electricity and magnetism 1864 Maxwell finds that light is an electromagnetic wave 1900 Planck solves the blackbody problem by assuming that photon energy is quantized as E = h F 1905 Einstein publishes the "photoelectric effect" experiment, providing the first direct measurement of photon energy and momentum. 1905 Theory of special relativity completed. 1913 Bohr model of the atom published 1924 de Broglie postulates that for particles with mass, Momentum * Wavelength = PlanckConstant 1927 Davisson and Germer experimentally verify the de Broglie relation for electrons.
l
Year c. 2400 Abacus invented by the Babylonians c. 500 First known use of the number "0" in Ancient India c. 300 Pingala develops binary numbers c. 100 Negative numbers used in Ancient China c. 200 Logarithms developed in Ancient India c. 600 Brahmagupta develops a placevalue number system c. 1400 Kerala school of astronomy and mathematics in South India invents the floating point number system. 1585 Stevin popularizes decimal numbers in Europe 1614 Napier develops logarithm tables 1622 Oughtred develops the slide rule 1642 Pascal builds a mechanical calculator 1671 Leibniz builds a mechanical calculator 1910 Vacuum tube invented 1646 ENIAC built. 17468 vacuum tubes. 5000 additions per second. 1947 Transistor invented 1957 Fortran compiler developed. This was the first highlevel programming language 1967 Dijkstra declares that the "goto" statement is harmful and advocates structured programming
The world's fastest supercomputer is the Tianhe2, built in 2013. It has the following properties:
Speed 34 Petaflops Processor cores 384000 Memory 1357 Terabytes Disk drives 12.4 Petabytes Power 24 MWatts Footprint 700 meters^2
1624 Slide rule age begins when Wingate publishes a table of logarithms 1630 Oughtred builds a circular slide rule 1632 Oughtred develops the modern design for the slide rule 1677 First commercial slide rule 1976 Slide rule era ends when Texas Instruments creates the electronic calculator, available for $104 in 2015 dollars.
Q = Wheat produced per Hectare / 1000 kg 1 Hectare = 10000 meters^2 Energy required to feed 1 person for 1 year = 2000 Calories/day * 4.2e3 Joules/Calorie * 365 days/year = 3.1e9 Joules/year Energy produced by 1 Hectare of land per year = Q * 1000 kg * 4.5e6 Joules/kg = 4.5e9 Q Number of people that can be supported by 1 Hectare of land = 1.5 Q Population density (People/Hectare) World .53 India 3.8 U.K. 2.6 Germany 2.3 Italy 2.0 China 1.4 France 1.2 USA .32
500  1400 Medieval 1400  1600 Renaissance 1600  1760 Baroque Monteverdi, Vivaldi, Bach, Handel 1720  1770 Galant Gluck 1730  1820 Classical Mozart 1780  1910 Romantic Beethoven, Brahms, Wagner 1890  Now Modern Prokofiev, Shostakovich
1567 1643 Monteverdi 1637 1707 Buxtehude 1653 1706 Pachelbel 1659 1695 Purcell 1663 1713 Corelli 1671 1751 Albinoni 1678 1741 Vivaldi 1681 1767 Telemann 1685 1750 Bach 1685 1759 Handel 1732 1809 Haydn 1756 1791 Mozart 1770 1827 Beethoven 1782 1840 Paganini 1797 1828 Schubert 1803 1869 Berlioz 1809 1847 Mendelssohn 1810 1849 Chopin 1810 1856 Schumann 1813 1883 Wagner 1813 1901 Verdi 1833 1897 Brahms 1835 1921 SaintSaens 1838 1920 Bruch 1840 1893 Tchaikovsky 1841 1904 Dvorak 1858 1924 Puccini 1860 1911 Mahler 1862 1918 Debussy 1864 1949 Strauss 1865 1957 Sibelius 1891 1953 Prokofiev 1906 1975 Shostakovich 1873 1943 Rachmaninov 1882 1971 Stravinsky 1910 1981 Barber
Monteverdi L'Orfeo 1607 First opera Purcell Dido and Aeneas 1683 Handel Agrippina 1710 Handel Giulio Cesare 1724 Handel Theodora 1750 Gluck Orfeo ed Euridice 1762 Gluck Iphigenie en Tauride 1779 Mozart The Marriage of Figaro 1786 Mozart Don Giovanni 1787 Mozart The Magic Flute 1791 Beethoven Fidelio 1805 Rossini The Barber of Seville 1616 Rossini Othello 1816 Rossini The Thieving Magpie 1817 Rossini William Tell 1829 Wagner The Flying Dutchman 1843 Wagner Tannhauser 1845 Wagner Lohengrin 1850 Verdi Rigoletto 1851 Verdi The Troubadour 1853 Verdi La Traviata 1853 Offenbach Orpheus in the Underworld 1858 Berlioz Les Troyens 1858 Wagner Tristan and Isolde 1865 Verdi Don Carlos 1867 Wagner Das Rheingold 1869 Ring cycle 1 Wagner Die Walkure 1870 Ring cycle 2 Verdi Aida 1871 Strauss II Die Fledermaus 1874 Bizet Carmen 1875 Wagner Siegfried 1876 Ring cycle 3 Wagner Gotterdammerung 1876 Ring cycle 4 SaintSaens Samson and Delilah 1877 Tchaikovsky Eugene Onegin 1879 Offenbach The Tales of Hoffman 1881 Wagner Parsifal 1882 Delibes Lakme 1883 Verdi Otello 1887 Humperdinck Hansel and Gretal 1893 Puccini La Boheme 1896 Puccini Tosca 1900 Debussy Pelleas et Melisande 1902 Puccini Madama Butterfly 1904 Strauss Salome 1905 Strauss Elektra 1909 Prokofiev The Love for Three Oranges 1921 Puccini Turandot 1926 Britten Peter Grimes 1945 Bernstein Candide 1956
2500 An ensemble of lyres was played in the ancient city of Ur, including lyres, harps, flutes, and reed instruments. 1000 Bowed instruments first developed, such as the Lyre 1200 The guitar comes into use in Europe 1555 Amati develops the fourstring violin 1700 Cristofori develops the first piano, an instrument where the string is struck by a hammer. Early pianos had 5 octaves 1785 Tourte develops the modern bow 1810 Broadwood develops a 6octave piano 1820 Broadwood develops a 7octave piano 1821 Erard develops the doubleescapement mechanism for the piano, a device that permitted repeating a note even if the key had not yet risen to its maximum vertical position. This facilitated rapid playing of repeated notes. 1835 Tuba invented 1847 Boehm advances the design of the flute, including a switch from wood to metal 1931 Beauchamp builds the first electric guitar
A harpshichord string is plucked and a piano string is hammered.
A harpsichord can't vary its volume.
The strings in a piano exert a force of 20 tons.
The Sydney Town Hall Grand Organ has pipes that are 64 feet long, which corresponds to a frequency of 8.5 Hertz.
Invented the opthalmascope, an instrument for examining the inside of the eye.
Developed theories of eye focus, depth perception, color vision, and motion perception.
Invented the "Helmholtz resonator" for measuring the frequency spectrum of sound.
Discovered the shape of the oscillation of a violin string.
Demonstrated that different combinations of resonators could mimic vowel sounds.
Measured the speed of neurons.
Developed the principle of conservation of energy and demonstrated that it applies to mechanics, heat, light, electricity and magnetism.
Demonstrated that muscle metabolism conserves energy.
Invented the field of psychology with his student Wilhelm Wundt.
In 1863, Helmholtz published "On the Sensations of Tone", which became the standard reference for the next century.
Students: Max Planck, Heinrich Kayser, Eugen Goldstein, Wilhelm Wien, Arthur Konig, Henry Augustus Rowland, Albert A. Michelson, Wilhelm Wundt, Fernando Sanford and Michael I. Pupin.
Amati (15051577) lived in Cremona, Italy, and developed the first violins, violas, and cellos.
This violin, now at the Metropolitan Museum of Art, may have been part
of a set made for the marriage of Philip II of Spain to Elisabeth of Valois in
1559, which would make it one of the earliest known violins in existence.
1) Bach, Mozart, and some old Italian and English composers are my favorites in music. Beethoven considerably less  but certainly Schubert.
(2) It is impossible for me to say whether Bach or Mozart means more to me. In music I do not look for logic. I am quite intuitive on the whole and know no theories. I never like a work if I cannot intuitively grasp its inner unity (architecture).
(3) I always feel that Handel is good  even perfect  but that he has a certain shallowness. Beethoven is for me too dramatic and too personal.
(4) Schubert is one of my favorites because of his superlative ability to express emotion and his enormous powers of melodic invention. But in his larger works I am disturbed by a certain lack of architectonics.
(5) Schumann is attractive to me in his smaller works because of their originality and richness of feeling, but his lack of formal greatness prevents my full enjoyment. In Mendelssohn I perceive considerable talent but an indefinable lack of depth that often leads to banality.
(6) I find a few lieder and chamber works by Brahms truly signficant, also in their structure. But most of his works have for me no inner persuasiveness. I do not understand why it was necessary to write them.
(7) I admire Wagner's inventiveness, but I see his lack of architectural structure as decadence. Moreover, to me his musical personality is indescribably offensive so that for the most part I can listen to him only with disgust.
(8) I feel that [Richard] Strauss is gifted, but without inner truth and concerned only with outside effects. I cannot say that I care nothing for modern music in general. I feel that Debussy is delicately colorful but shows a poverty of structure. I cannot work up great enthusiasm for something of that sort.
1920 Commercial radio broadcasts begin 1934 Commercial television appears 1941 RCA begins broadcasting 525line television. Television becomes widespread 1954 First color television broadcast 1965 Half of broadcasts are in color 1972 All broadcasts are in color 2008 Digital television era begins. All television is digital by 2010
1585 Stevin introduces decimal numbers. (For example, writing 1/8 as 0.125) 1637 Cartesian geometry published by Fermat and Descartes 1684 Leibniz publishes The Calculus 1761 Lambert proves that Pi is irrational 1821 Cauchy publishes the "epsilondelta" definition of a limit, which brought rigor to The Calculus. 1830 Galois publishes "Galois Theory", which explains why a general polynomial equation of order n can be solved in terms of radicals only if n <= 4. 1844 Louisville proves the existence of transcendental numbers 1851 Louisville constructs the first transcendental number 1854 Riemann publishes the Riemann Integral, the first rigorous definition of an integral. 1859 Riemann Hypothesis published 1860 Grassmann studies the question of the axiomatization of arithmetic. 1870 Heine defines "uniform continuity" 1872 Heine proves that a continuous function on an open interval need not be uniformly continuous. 1872 Weierstrass publishes the "Weierstrass function", the first example of a function that is continuous everywhere but differentiable nowhere. 1873 Hermite proves that "e" is transcendental 1874 Cantor proves that the algebraic numbers are countable and that the real numbers are uncountable, using the "diagonal slash" argument. 1874 Cantor publishes the first attempt at a rigorous set theory. 1878 Cantor proves that the transcendental numbers and the real numbers have the same cardinality, thus estabilishing the ubiquity of transcendental numbers 1878 Cantor publishes the "Continuum Hypothesis": "There is no set whose cardinality is strictly between that of the integers and the real numbers." In 1900, Hilbert included the question of the Continuum Hypothesis in his list of 23 unsolved problems. 1882 Lindemann proves that Pi is transcendental. A corollary is the imposibility of squaring a circle with a compass and straightedge. 1883 Cantor publishes the Cantor Set, a rich source of counterexamples 1887 Poincare discovers the phenomenon of "Chaos" while studying celestial mechanics. There exist orbits that are neither unbounded nor limiting to a stable state. 1889 Peano publishes a set of axioms for arithmetic which are now the standard. 1898 Hadamard defines a dynamical system where all orbits exponential diverge from each other with a positive Lyapunov exponent. 1900 Hilbert publishes a list of 23 unsolved problems. They include: The Continuum Hypothesis (proved independent of ZFC by Godel) Prove that the axioms of arithmetic are consistent. (proved impossible by Godel) The Riemann Hypothesis (still unresolved) What is the densest sphere packing? (resolved in 1998) 1901 Russell publishes "Russell's Paradox", which shows that Cantor's set theory leads to a contradiction. This was resolved in 1922 by the ZermeloFraenkel axioms of set theory. 1902 Lebesgue publishes the Lebesque Integral, a generalization of the Riemann Integral. "Lebesgue Measure" is the standard way of assigning a measure to subsets of ndimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. The Lebesgue measure of the set of rational numbers in the interval [0,1] is 0, and the real numbers on this interval have measure 1. The Cantor set is an example of an uncountable set that has Lebesgue measure zero. 1904 Poincare Conjecture published 1904 Zermelo defines the Axiom of Choice. Previously, mathematicians had been using this axiom implicitly without realizing it. Kronecker's opposition to Cantor's theories became the inspiration for the mathematical outlook of "Constructivism", which asserts that it is necessary to construct a mathematical object to prove that it exists (proving its nonexistence does not imply its existence). Constructivism is at odds with the Axiom of Choice and the Law of the Excluded Middle. 1922 ZermeloFraenkel axioms of set theory developed (ZF). This resolved Russell's Paradox. 1924 BanachTarsky paradox published, exhibiting a spooky consequence of the Axiom of Choice. 1931 Godel proves the Incompleteness Theorems. For any set of axioms that are nontrivial and consistent, there will exist statements about the natural numbers that are true but cannot be proven within the system. Also, the system cannot prove its own consistency. Cantor's "diagonal slash" argument was an inspiration for these theorems. 1935 Bourbaki textbooks published, with the aim of reformulating mathematics on an extremely abstract and formal but selfcontained basis. With the goal of grounding all of mathematics on set theory, the authors strove for rigour and generality. 1940 Godel proves that the Axiom of Choice and the Continuum Hypothesis cannot be disproved with the ZermeloFraenkel axioms (ZF). He also established that the Continuum Hypothesis cannot be disproved even if the Axiom of Choice is added to the ZermeloFraenkel axioms (ZFC). 1961 Lorenz finds that computer simulations of weather have extreme sensitivity to initial conditions. 1963 Cohen proves that the Axiom of Choice and the Continuum Hypothesis cannot be proved with the ZermeloFraenkel axioms, establishing that they are independent of ZF. 1967 Mandelbrot publishes examples of fractals from nature 1967 Bishop publishes "Foundations of Constructive Analysis", where he proved most of the important theorems in real analysis by constructive methods. 1982 Mandelbrot publishes "The Fractal Geometry of Nature" 1983 Langlands Program published 1994 Wiles proves Fermat's Last Theorem 2000 Millenium Prize problems published. They include: The Riemann Hypothesis P versus NP The Poincare Conjecture Navierâ€“Stokes existence and smoothness 2002 Perelman proves the Poincare Conjecture 2009 Chau proves the Fundamental Lemma for the Langlands Program 2013 Zhang and Maynard publish results that constitute progress toward resolving the twin prime conjecture.
f(1) = 0 f(2) = 1 f(3) = 1/2 f(4) = 1/3 f(5) = 2/3 f(6) = 1/4 f(7) = 3/4 f(8) = 1/5 f(9) = 2/5 f(10) = 3/5 f(11) = 4/5 f(12) = 1/6 f(13) = 5/6 f(14) = 1/7 f(15) = 2/7 f(16) = 3/7 etc.Every rational number corresponds to a unique integer and every integer corresponds to a unique rational number (a "bijection").
If a set can be bijected with the integers we say it is "Countable". The rational numbers are countable.
"Algebraic numbers" are numbers that can be expressed as the root of a nonzero ndegree polynomial with integer coefficients. The rational numbers correspond to roots of polynomials of degree 1. The algebraic numbers are countable.
Rational number Expressible as A/B, where A and B are integers. Irrational number Not a rational number. Algebraic number Expressible as the root of a nonzero polynomial with integer coefficients. Transcendental number Not an algebraic number.Suppose we attenpt to count the real numbers on the interval [0,1]. Let X=f(I) be a bijection between the positive integers I and the reals X, where every real number X is represented by some integer I. Let g(X,n) be the nth digit of X to the right of the decimal point. Define a number Z such that g(Z,n) = g(f(n),n). Z is not equal to f(I) for any integer I, and so Z is not present in the counting. Any attempt to count the reals will result in at least one missed number, hence the reals are uncountable. We say that the integers are "countably infinite" and the reals are "uncountably infinite".
This is Cantor's "diagonal slash" argument that he used to establish that the real numbers are more numerous than the integers. Godel's theorems are inspired by the diagonal slash argument.
In terms of subsets,
Integers < Rational numbers < Algebraic numbers < Transcendental numbersA countable set has Lebesgue measure zero. In terms of Lebesgue measure over the interval [0,1],
0 = Rational numbers = Algebraic numbers < Transcendental numbers = Real numbers = 1Two sets have the same "cardinality" if and only if a bijection exists between them. In terms of cardinality,
Integers = Rational numbers = Algebraic numbers < Transcendental numbers = Real numbersIn terms of cardinality, real numbers are infinitely more numerous than algebraic numbers.
The Continuum Hypothesis conjectures that there exists no set whose cardinality is strictly between that of the integers and the real numbers. If such a set S existed, its cardinality would be such that
Integers < S < Real numbers
Hermann Weyl, 1949: "Mathematics with Brouwer gains its highest intuitive clarity. He succeeds in developing the beginnings of analysis in a natural manner, all the time preserving the contact with intuition much more closely than had been done before. It cannot be denied, however, that in advancing to higher and more general theories the inapplicability of the simple laws of classical logic eventually results in an almost unbearable awkwardness. And the mathematician watches with pain the greater part of his towering edifice which he believed to be built of concrete blocks dissolve into mist before his eyes."
Hermann Weyl, 1939: "In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain."
Poincare: "There is no actual infinite; the Cantorians have forgotten this, and that is why they have fallen into contradiction."
21512155 Enterprise Seasons 14 (Captain Archer) 22662269 Star Trek Seasons 13 (Captain Kirk) 23642370 The Next Generation Seasons 17 (Captain Picard) 23692375 Deep Space 9 Seasons 17 (Captain Sisqo) 23712404 Voyager Seasons 17 (Captain Janeway) 2700 Vulcan is in a violent colonial era 900 Vulcans develop space travel 400 On Vulcan, Surak introduces logic. Subsequent wars force some Vulcans to flee Vulcan and colonize Romulus, and they become the Romulans. 930 Kahless introduces the concept of honor to a warlike Klingon society 1900 Vulcan civilization stabilizes and interstellar exploration begins 1947 Vulcans develop warp drive 1947 Klingons develop warp drive, possibly from reverseengineering 2047 Vulcans warp drives reach a speed of Warp 2 2057 Earth launches its first spacecraft, Sputnik. Vulcan begins monitoring the Earth 20492053 Earth WW III 2050 War begins between Vulcan and Andoria 2063 On Earth, Zefrim Cochrane invents the Warp drive, prompting the Vulcans to make official contact with Earth 2151 Klingons encounter Earthlings 21512152 Enterprise Season 1 (Captain Archer) 21542155 Enterprise Season 4 21562160 EarthRomulan War 2258 Star Trek 11. Timeline forks here to the alternate timeline of Star trek 12. 2060 Battle of Cheron. The Federation, Klingons, Andorians, and Tellarites defeat the Romulans. Neutral zone established along the Romulan border. Romulans go into isolation. 2152 Skirmish between a Federation and Romulan warship 2161 Founding of the United Federation of Planets, including Earth, Vulcan, and Andoria. Enterprise season finale 22662267 Star Trek Season 1 (Captain Kirk) 2266 Romulans end their isolation and attacks the Federation 22682269 Star Trek Season 3 2271 Klingons defeat the Romulans at the battle of Klach D'Kel Brakt 2285 Wrath of Khan 2285 Klingons acquire cloaking technology from the Romulans 2311 Romulan empire goes into isolation 2344 The Federation and the Klingon Empire negotiate a peace treaty 2347 FederatonCardassian war begins 2353 The Borg become aware of the existence of humanity 2353 The Federation and the Klingon Empire form an alliance 2354 The U.S.S. Raven is sent to investigate the Borg and disappears 2364 The Next Generation Season 1 (Captain Picard) 2364 Several Federation and Romulan outposts are destroyed by the Borg, prompting the Romulan Empire to come out of isolation and negotiate with the Federation. 2364 The Federation makes first contact with the Ferengi 2366 The Borg invade the Alpha Quadrant and attack the Federation 2367 Gowron becomes Chancellor of the Klingon Empire 2367 The Federaton and the Cardassian Union sign a peace treaty 2369 Cardassian occupation of Bajor ends 2369 Starfleet takes posession of Deep Space 9 2369 Bajor wormhole discovered. Starfleet begins exploring the Gamma Quadrant 2369 Deep Space 9 Season 1 (Captain Sisqo) 2370 The Next Generation Season 7 2370 The Federation encounters the Dominion in the Gamma Quadrant, in which the U.S.S. Odyssey is destroyed by a Jem'Hedar ship 2371 The Federation and the Romulans begin cooperating in their struggle against the Dominion. The Romulans give the Federation cloaking technology 2371 Voyager Season 1 (Captain Janeway) 2373 The Dominion invades the Alpha Quadrant and attacks the Federation the Klingons, and the Cardassians. The Cardassian homeworld is captured. 2374 The Romulans enter the war against the Dominion 2375 The Federation, Klingons, Romulans, and Cardassians defeat the Dominion and eject them from the Alpha Quadrant 2375 Deep Space 9 Season 7 2387 Romulus destroyed by a supernova 2404 Voyager Season 7 Warp factor Speed in light units 1 1 2 10 3 39 4 102 5 214 6 392 7 656 8 1024 9 1516 10 Infinite