Cosmos
by Carl Sagan · 1 Jan 1980 · 404pp · 131,034 words
greatest metropolis of the age, the Egyptian city of Alexandria. Here there lived a man named Eratosthenes. One of his envious contemporaries called him “Beta,” the second letter of the Greek alphabet, because, he said, Eratosthenes was second best in the world in everything. But it seems clear that in almost everything
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Eratosthenes was “Alpha.” He was an astronomer, historian, geographer, philosopher, poet, theater critic and mathematician. The titles of
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might easily have ignored. Sticks, shadows, reflections in wells, the position of the Sun—of what possible importance could such simple everyday matters be? But Eratosthenes was a scientist, and his musings on these commonplaces changed the world; in a way, they made the world
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mind to do an experiment, actually to observe whether in Alexandria vertical sticks cast shadows near noon on June 21. And, he discovered, sticks do. Eratosthenes asked himself how, at the same moment, a stick in Syene could cast no shadow and a stick in Alexandria, far to the north, could
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at an angle of seven degrees. Seven degrees is something like one-fiftieth of three hundred and sixty degrees, the full circumference of the Earth. Eratosthenes knew that the distance between Alexandria and Syene was approximately 800 kilometers, because he hired a man to pace it out. Eight hundred kilometers times
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50 is 40,000 kilometers: so that must be the circumference of the Earth.* This is the right answer. Eratosthenes’ only tools were sticks, eyes, feet and brains, plus a taste for experiment. With them he deduced the circumference of the Earth with an error
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be tempted to make voyages of exploration, to seek out undiscovered lands, perhaps even to attempt to sail around the planet? Four hundred years before Eratosthenes, Africa had been circumnavigated by a Phoenician fleet in the employ of the Egyptian Pharaoh Necho. They set sail, probably in frail open boats, from
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alternate interior angles are equal”), angle B equals angle A. So by measuring the shadow length in Alexandria, Eratosthenes concluded that Syene was A = B = 7° away on the circumference of the Earth. After Eratosthenes’ discovery, many great voyages were attempted by brave and venturesome sailors. Their ships were tiny. They had
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an unexplored ocean. The stars are the friends of explorers, then with seagoing ships on Earth and now with spacefaring ships in the sky. After Eratosthenes, some may have tried, but not until the time of Magellan did anyone succeed in circumnavigating the Earth. What tales of daring and adventure must
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earlier have been recounted as sailors and navigators, practical men of the world, gambled their lives on the mathematics of a scientist from Alexandria? In Eratosthenes’ time, globes were constructed portraying the Earth as viewed from space; they were essentially correct in the well-explored Mediterranean but became more and more
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say they have been prevented by an opposing continent, for the sea remained perfectly open, but, rather, through want of resolution and scarcity of provision.… Eratosthenes says that if the extent of the Atlantic Ocean were not an obstacle, we might easily pass by sea from Iberia to India.… It is
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following few centuries, which completed the geographical exploration of the Earth. Columbus’ first voyage is connected in the most straightforward way with the calculations of Eratosthenes. Columbus was fascinated by what he called “the Enterprise of the Indies,” a project to reach Japan, China and India not by following the coastline
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of Africa and sailing East but rather by plunging boldly into the unknown Western ocean—or, as Eratosthenes had said with startling prescience, “to pass by sea from Iberia to India.” Columbus had been an itinerant peddler of old maps and an assiduous
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reader of the books by and about the ancient geographers, including Eratosthenes, Strabo and Ptolemy. But for the Enterprise of the Indies to work, for ships and crews to survive the long voyage, the Earth had to
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be smaller than Eratosthenes had said. Columbus therefore cheated on his calculations, as the examining faculty of the University of Salamanca quite correctly pointed out. He used the smallest
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the outlines of its continents, confirming that many of the ancient mapmakers were remarkably competent. What a pleasure such a view would have given to Eratosthenes and the other Alexandrian geographers. It was in Alexandria, during the six hundred years beginning around 300 B.C., that human beings, in an important
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of age. Genius flourished there. The Alexandrian Library is where we humans first collected, seriously and systematically, the knowledge of the world. In addition to Eratosthenes, there was the astronomer Hipparchus, who mapped the constellations and estimated the brightness of the stars; Euclid, who brilliantly systematized geometry and told his king
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last years was a concordance and calibration of the chronologies of ancient civilizations, very much in the tradition of the ancient historians Manetho, Strabo and Eratosthenes. In his last, posthumous work, “The Chronology of Ancient Kingdoms Amended,” we find repeated astronomical calibrations of historical events; an architectural reconstruction of the Temple
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five meters long. His discoveries with the telescope would by themselves have ensured his place in the history of human accomplishment. In the footsteps of Eratosthenes, he was the first person to measure the size of another planet. He was also the first to speculate that Venus is completely covered with
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. It is hardly anything at all. On the other hand, suppose our time traveler had persuaded Queen Isabella that Columbus’ geography was faulty, that from Eratosthenes’ estimate of the circumference of the Earth, Columbus could never reach Asia. Almost certainly some other European would have come along within a few decades
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of knowledge. The Ptolemys did not merely collect established knowledge; they encouraged and financed scientific research and so generated new knowledge. The results were amazing: Eratosthenes accurately calculated the size of the Earth, mapped it, and argued that India could be reached by sailing westward from Spain. Hipparchus anticipated that stars
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poignant lost opportunity for the human species. We have in this book devoted attention to some of our ancestors whose names have not been lost: Eratosthenes, Democritus, Aristarchus, Hypatia, Leonardo, Kepler, Newton, Huygens, Champollion, Humason, Goddard, Einstein—all from Western culture because the emerging scientific civilization on our planet is mainly
Big Bang
by Simon Singh · 1 Jan 2004 · 492pp · 149,259 words
would everybody else on the globe, even if they lived down under. The feat of measuring the size of the Earth was first accomplished by Eratosthenes, born in about 276 BC in Cyrene, in modern-day Libya. Even when he was a little boy it was clear that
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Eratosthenes had a brilliant mind, one that he could turn to any discipline, from poetry to geography. He was even nicknamed Pentathlos, meaning an athlete who
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participates in the five events of the pentathlon, hinting at the breadth of his talents. Eratosthenes spent many years as the chief librarian at Alexandria, arguably the most prestigious academic post in the ancient world. Cosmopolitan Alexandria had taken over from
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books and whispering to each other, because this was a vibrant and exciting place, full of inspiring scholars and dazzling students. While at the library, Eratosthenes learned of a well with remarkable properties, situated near the town of Syene in southern Egypt, near modern-day Aswan. At noon on 21 June
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each year, the day of the summer solstice, the Sun shone directly into the well and illuminated it all the way to the bottom. Eratosthenes realised that on that particular day the Sun must be directly overhead, something that never happened in Alexandria, which was several hundred kilometres north of
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can appear overhead. Aware that the Earth’s curvature was the reason why the Sun could not be overhead at both Syene and Alexandria simultaneously, Eratosthenes wondered if he could exploit this to measure the circumference of the Earth. He would not necessarily have thought about the problem in the same
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the Sun hit the Earth at noon on 21 June. At exactly the same moment that sunlight was plunging straight down the well at Syene, Eratosthenes stuck a stick vertically in the ground at Alexandria and measured the angle between the Sun’s rays and the stick. Crucially, this angle is
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between two radial lines drawn from Alexandria and Syene to the centre of the Earth. He measured the angle to be 7.2°. Figure 1 Eratosthenes used the shadow cast by a stick at Alexandria to calculate the circumference of the Earth. He conducted the experiment at the summer solstice, when
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the distance between Syene and Alexandria represents 7.2/360, or 1/50 of the Earth’s circumference. The rest of the calculation is straightforward. Eratosthenes measured the distance between the two towns, which turned out to be 5,000 stades. If this represents 1/50 of the total circumference of
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the circumference of the Earth would be 46,250 km, which is only 15% bigger than the actual value of 40,100 km. In fact, Eratosthenes may have been even more accurate. The Egyptian stade differed from the Olympic stade and was equal to just 157 metres, which gives a circumference
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of 39,250 km, accurate to 2%. Whether he was accurate to 2% or 15% is irrelevant. The important point is that Eratosthenes had worked out how to reckon the size of the Earth scientifically. Any inaccuracy was merely the result of poor angular measurement, an error in
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the Syene—Alexandria distance, the timing of noon on the solstice, and the fact that Alexandria was not quite due north of Syene. Before Eratosthenes, nobody knew if the circumference was 4,000 km or 4,000,000,000 km, so nailing it down to roughly 40,000 km was
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a stick and a brain. In other words, couple an intellect with some experimental apparatus and almost anything seems achievable. It was now possible for Eratosthenes to deduce the size of the Moon and the Sun, and their distances from the Earth. Much of the groundwork had already been laid by
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earlier natural philosophers, but their calculations were incomplete until the size of the Earth had been established, and now Eratosthenes had the missing value. For example, by comparing the size of the Earth’s shadow cast upon the Moon during a lunar eclipse, as shown
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in Figure 2, it was possible to deduce that the Moon’s diameter was about one-quarter of the Earth’s. Once Eratosthenes had shown that the Earth’s circumference was 40,000 km, then its diameter was roughly (40,000 ÷ π) km, which is roughly 12,700
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an indication of the Earth’s diameter. The Earth’s diameter is therefore roughly four times the Moon’s diameter. It was then easy for Eratosthenes to estimate the distance to the Moon. One way would have been to stare up at the full Moon, close one eye and stretch out
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of 320,000 km. Next, thanks to a hypothesis by Anaxagoras of Clazomenae and a clever argument by Aristarchus of Samos, it was possible for Eratosthenes to calculate the size of the Sun and how far away it was. Anaxagoras was a radical thinker in the fifth century BC who deemed
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Moon, except that now the Moon has taken the place of our fingernail as an object of known size and distance. The amazing achievements of Eratosthenes, Aristarchus and Anaxagoras illustrate the advances in scientific thinking that were taking place in ancient Greece, because their measurements of the universe relied on logic
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being transformed into another. They had no inkling of the underlying chemical or biochemical mechanisms at work. So, the Egyptians were technologists, not scientists, whereas Eratosthenes and his colleagues were scientists, not technologists. The intentions of the Greek scientists were identical to those described two thousand years later by Henri Poincaré
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distance to the Moon, which depends on knowing the diameter of the Moon, which depends on knowing the diameter of the Earth, and that was Eratosthenes’ great breakthrough. These distance and diameter stepping stones were made possible by exploiting a deep vertical well on the Tropic of Cancer, the Earth’s
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view of the world in which the Earth was a central static globe with the universe revolving around it. Table 1 The measurements made by Eratosthenes, Aristarchus and Anaxagoras were inaccurate, so the table below corrects previously quoted figures by providing modern values for the various distances and diameters. Earth’s
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a factor of ten, and we now know that the Milky Way is about 100,000 light years across and 10,000 light years thick. Eratosthenes had been shocked when he measured the distance to the Sun, and Bessel had been staggered by the distance to the nearest stars, but the
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ancient Greeks dismissing the possibility of measuring the size of the Earth or the distance to the Sun. However, the first generation of scientists, including Eratosthenes and Anaxagoras, invented techniques that allowed them to span the globe and the Solar System. Then Herschel and Bessel used brightness and parallax to measure
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; radioactive 285—6; see also atomic abundances ellipses 54-6,55,58,119 epicycles and deferents 30, 31, 32,45, 58, 65, 66,124, 367 Eratosthenes 11-16, 12,17,19,20,177, 195 Eta Aquilae 198-9 ether 93-8, 96,101-3,141-2,145,306, 368 evolution 77
To Explain the World: The Discovery of Modern Science
by Steven Weinberg · 17 Feb 2015 · 532pp · 133,143 words
anticipation of later ideas of impetus or momentum. But there were no more creative scientists or mathematicians of the caliber of Eudoxus, Aristarchus, Hipparchus, Euclid, Eratosthenes, Archimedes, Apollonius, Hero, or Ptolemy. Whether or not because of the rise of Christianity, soon even the commentators disappeared. Hypatia was killed in 415 by
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Sun as multiples of the size of the Earth. The size of the Earth was measured a few decades after the work of Aristarchus by Eratosthenes. Eratosthenes was born in 273 BC at Cyrene, a Greek city on the Mediterranean coast of today’s Libya, founded around 630 BC, that had become
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unfortunately disappeared, but were widely quoted in antiquity. The measurement of the size of the Earth by Eratosthenes was described by the Stoic philosopher Cleomedes in On the Heavens,16 sometime after 50 BC. Eratosthenes started with the observations that at noon at the summer solstice the Sun is directly overhead at
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Syene, an Egyptian city that Eratosthenes supposed to be due south of Alexandria, while measurements with a gnomon at Alexandria showed the noon Sun at the solstice to be one-fiftieth
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of the Earth must be 250,000 stadia. How good was this estimate? We don’t know the length of the stadion as used by Eratosthenes, and Cleomedes probably didn’t know it either, since (unlike our mile or kilometer) it had never been given a standard definition. But without knowing
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the length of the stadion, we can judge the accuracy of Eratosthenes’ use of astronomy. The Earth’s circumference is actually 47.9 times the distance from Alexandria to Syene (modern Aswan), so the conclusion of
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Eratosthenes that the Earth’s circumference is 50 times the distance from Alexandria to Syene was actually quite accurate, whatever the length of the stadion.* In
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his use of astronomy, if not of geography, Eratosthenes had done quite well. 8 The Problem of the Planets The Sun and Moon are not alone in moving from west to east through the
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his own version of the planetary scheme described in Ptolemy’s Planetary Hypotheses. It was a major occupation of this Baghdad group to improve on Eratosthenes’ measurement of the size of the Earth. Al-Farghani in particular reported a smaller circumference, which centuries later encouraged Columbus (as mentioned in an earlier
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the Earth. (The actual value is about 11,600.) Measuring the diameter of the Earth was the next task. 12. The Size of the Earth Eratosthenes used the observation that at noon on the summer solstice, the Sun at Alexandria is 1/50 of a full circle (that is, 360°/50
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to Alexandria must also be 7.2°, or 1/50 of a full circle. (See Figure 6.) Hence on the basis of the assumptions of Eratosthenes, the Earth’s circumference must be 50 times the distance from Alexandria to Syene. Figure 6. The observation used by
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Eratosthenes to calculate the size of the Earth. The horizontal lines marked with arrows indicate rays of sunlight at the summer solstice. The dotted lines run
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, 149–50 distance to stars, 70 distance to Sun, 63, 66–68, 70, 75, 83, 90, 164, 295–301 epicycles of planets and, 303–7 Eratosthenes on, 75–76, 301–2 Galileo on, 184–88 Greeks on motion of, 10, 70–72, 79–86, 89, 153–54 Heraclides on, 89 medieval
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, 92–95, 117, 151–52, 166, 169, 254, 323–28 equinoxes, 58, 60–61 precession of, 74–75, 107, 118, 153, 241–42, 244, 248 Eratosthenes, 51, 75–76, 107, 301–2 ether, 10, 258 Euclid, 15, 17–19, 35, 37, 47, 51, 69, 105, 119, 126, 206, 210, 223, 232
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Epicurus, 22, 46–47 Gassendi, Pierre, 46, 234 Geminus, 95–97, 157, 246 general theory of relativity, 14, 234, 250–53 genetics, 266 Geographic Memoirs (Eratosthenes), 75 Geometrie (Descartes), 223 geometry, xiv, 125, 139, 197, 199 algebra vs., 40–41 analytic, 40, 205–7 Galileo and, 40–41 Greeks and, 4
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, 142 Henry VII, king of England, 253 Heraclides, 85–86, 89, 124, 134, 153, 303 Heraclitus, 6, 8, 13, 57 Heraclius, Byzantine emperor, 103 Hermes (Eratosthenes), 75 Herodotus, 45–46, 58 Hero of Alexandria, 36–37, 41, 51, 137, 189, 208, 210, 289–91, 348 Hertz, Heinrich, 259 Hesiod, 13, 47
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and the Earth (Oresme), 135 On the Heavens (Aristotle), 25, 27, 64, 80, 127 On the Heavens (Cleomedes), 75 On the Measurement of the Earth (Eratosthenes), 75 On the Motion of Bodies in Orbit (Newton), 231 On the Nature of Things (Lucretius), 46 On the Republic (Cicero), 17, 71 On the
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vernal equinox. * On the basis of his own observations of the star Regulus, Ptolemy in Almagest gave a figure of 1° in approximately 100 years. * Eratosthenes was lucky. Syene is not precisely due south of Alexandria (its longitude is 32.9° E, while that of Alexandria is 29.9° E) and
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noon Sun at the summer solstice is not precisely overhead at Syene, but about 0.4° from the vertical. The two errors partly cancel. What Eratosthenes had really measured was the ratio of the circumference of the Earth to the distance from Alexandria to the Tropic of Cancer (called the summer
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.7° = 46.75 times greater than the distance between Alexandria and the Tropic of Cancer, just a little less than the ratio 50 given by Eratosthenes. * For the sake of clarity, when I refer to planets in this chapter, I will mean just the five: Mercury, Venus, Mars, Jupiter, and Saturn
Accessory to War: The Unspoken Alliance Between Astrophysics and the Military
by Neil Degrasse Tyson and Avis Lang · 10 Sep 2018 · 745pp · 207,187 words
equator and a prime meridian at right angles to it. With its main parallel and its prime meridian crossing at the Aegean island of Rhodes, Eratosthenes’s ancient world map had a grid that Hipparchus found arbitrary. Ptolemy’s map, with its prime meridian passing through the westernmost known islands in
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disbelieving Strabo. 20.Roseman, Pytheas, 7–20, writes that eighteen known ancient writers referred to Pytheas by name between 300 BC and AD 550, notably Eratosthenes, Hipparkhos, Polybius, Strabo, and Pliny the Elder. Two more—Poseidonios and Diodoros—likely used his information but did not name him in their extant works
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). 47.Parry, Age of Reconnaissance, 69–70; Williams, Sails to Satellites, 9, 16, 18; Randles, “Evaluation of Columbus’ ‘India’ Project,” 54–55. Seventeen centuries earlier, Eratosthenes had raised the idea of heading west from Lisbon to reach China. For Mandeville, see C. W. R. D. Moseley, “Behaim’s Globe and ‘Mandeville
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–9, 443–44n Enlargement of the Committee on the Peaceful Uses of Outer Space, 502n Enola Gay, 457n entanglement (photons), 313, 351 ephemerides, 82, 94 Eratosthenes, 87, 434n, 437n Euclid, 44, 107 Eudoxus of Cnidus, 71, 72, 73 European Defence Agency, 327 European Geostationary Navigation Overlay Service (EGNOS), 328 European Launcher
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maneuverable satellites, 397, 531n Manhattan Project, 390, 401 Mansfield Amendment, 222–23 Mao Zedong, 318, 351 maps and mapmaking at end of thirteenth century, 78 Eratosthenes, 87 first extant terrestrial globe (“Erdapfel”), 87, 436n first maps of Earth’s inhabited regions, 70–71 as a form of political and social power
Mapmatics: How We Navigate the World Through Numbers
by Paulina Rowinska · 5 Jun 2024 · 361pp · 100,834 words
the Earth. The estimates were almost two millennia old. Here comes the sun Born in 276 bce in Cyrene in Ancient Greece (today’s Libya), Eratosthenes was a successful mathematician, geographer, poet, astronomer and music theorist. He moved to Egypt to become the chief librarian of the Library of Alexandria, one
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this result, didn’t survive, it was described a few centuries later (we aren’t sure of the exact time) by the Greek astronomer Clomedes. Eratosthenes thought about the Earth as a sphere. He knew that on the day of the summer solstice, the longest day of the year in the
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the sun is at its highest point in the sky – a deep well in Syene (today’s Aswan in Egypt) was lit by direct sunlight, Eratosthenes wanted to understand the position of the sun at the same time in Alexandria. By a lucky coincidence, the two cities lay along the same
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meridian, so the local noon occurred in both places at the same time.* To find the angle between sun rays and the Earth in Alexandria, Eratosthenes measured the angle between a vertical rod called a gnomon and its shadow, which turned out to be one-fiftieth of the full circle (360
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Alexandria and Syene had already been measured to be 5,000 stadia, which is an ancient unit of measurement. To estimate the Earth’s circumference, Eratosthenes was missing just the central angle between the two cities. FIGURE 1.1: At the local noon of the summer solstice, sun rays are perpendicular
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shadow in Alexandria. The central angle between Syene and Alexandria and the angle between the sun rays and the gnomon in Alexandria have equal measures. Eratosthenes assumed that sun rays are parallel to each other, which, although not technically true, for all practical purposes is a reasonable assumption. The sun is
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radius in Alexandria) crossing two parallel lines (the extended sun rays in Alexandria and Syene). From an old theorem, still taught in geometry classes today, Eratosthenes knew that this is a pair of equal angles.* This meant that the central angle between the two cities was equal to one-fiftieth of
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: 50 × 5,000 = 250,000 stadia. Historians disagree on the exact definition of the stadium measurement, which makes it impossible to assess the accuracy of Eratosthenes’s estimate. That exact value aside, most researchers still agree that he got astonishingly close to the actual value of about 40,000 kilometres. He
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Alexandria were on the same meridian, the latter lay to the west. However, mathematical models never perfectly reflect reality. What’s most important is that Eratosthenes’s method was scientifically sound and, had he had access to more accurate measurement tools, his estimate wouldn’t have differed from the current best
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to the scholar Ferdinand Columbus, who happened to be Christopher’s son, the explorer was familiar with the work of ancient and medieval geographers, including Eratosthenes. He used this knowledge to persuade others of his idea to get to the Indies by travelling westward. However, aware that any reasonable potential funder
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’s talent for data fudging didn’t disappoint. Before Columbus, multiple scholars had attempted to measure the Earth’s circumference. As we’ve seen with Eratosthenes, who did a great job with the tools at his disposal, even the best methods didn’t guarantee a perfect estimate, and a big issue
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it to measure the size of the Earth. Although he described his project in a book aptly named Eratosthenes Batavus (which translates to The Dutch Eratosthenes), his method differed from the ancient measurement by Eratosthenes. Snell used triangulation to find the accurate distance between Alkmaar and another Dutch city, Bergen-op-Zoom, about
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Triangulation’, The Renaissance Mathematicus blogpost, 25 May 2012, https://thonyc.wordpress.com/2012/05/25/mapping-the-history-of-triangulation/. which translates to The Dutch Eratosthenes: Fokko Jan Dijksterhuis, ‘The Mutual Making of Sciences and Humanities: Willebrord Snellius, Jacob Golius, and the Early Modern Entanglement of Mathematics and Philology’, in The
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Geodetic Parameter Dataset ref1, ref2n equal-area (homolographic) projections ref1, ref2 Equator ref1, ref2, ref3, ref4, ref5, ref6, ref7, ref8, ref9 equidistant projections ref1 Eratosthenes ref1, ref2 Eratosthenes Batavus (Snell) ref1 ‘Error detecting and error correcting codes’ (Hamming) ref1 Esselstyn, Blake ref1 Euclid of Alexandria ref1 Euclidean metrics ref1, ref2, ref3, ref4
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, ref2 Oldham, Richard Dixon ref1 On-Road Integrated Optimization and Navigation (ORION) (UPS) ref1 On the Heavens (Aristotle) ref1 On the Measure of the Earth ( Eratosthenes) ref1 On the Mode of Communication of Cholera (Snow) ref1, ref2 ‘Orthogonal Map of the World’ (Peters) ref1 orthographic projections ref1 Osipovitch, Col. Gennadi ref1
Pathfinders: The Golden Age of Arabic Science
by Jim Al-Khalili · 28 Sep 2010 · 467pp · 114,570 words
for its size, another Greek scholar decided he could go one better than educated guesswork. He believed he could actually measure it. His name was Eratosthenes (c. 275–195 BCE) and he was the chief librarian of Alexandria, as well as being a brilliant astronomer and mathematician. His method for working
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the modern measurement of 24,900 miles that it would seem churlish and pedantic to find any fault with it. But the truth is that Eratosthenes was very lucky to have got so close. There were a number of serious errors, inaccuracies and crude guesses involved in his method that conspired
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. In any case, the fact that the number of paces came to exactly 5,000 stadia is suspicious and most modern historians do not believe Eratosthenes ever did have the distance measured in this way but had unwittingly used instead a value for the distance that itself had been calculated from
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so we move forward in time a thousand years to Abbāsid Baghdad and the band of astronomers working for al-Ma’mūn. They knew about Eratosthenes’ method from the writings of Ptolemy. In fact, Ptolemy quoted a later, revised but incorrect value for the circumference of the earth of just 180
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by 360 gives a figure of 24,500 miles, which is a more reliable figure than the one arrived at a thousand years earlier by Eratosthenes. Good scientist that he was, al-Ma’mūn then commissioned another expedition to carry out a second measurement, this time in the Syrian desert. Starting
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trying to credit those who arrived at the closest value. Al-Ma’mūn’s astronomers will have had to contend with the same issues as Eratosthenes. For instance, al-Raqqah is in fact about 1.5 degrees of latitude north of Palmyra as well as being about a degree of longitude
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(these explorers were unaware of the difference between a Roman and Arabic mile), they were often unwittingly quoting al-Farghāni quoting Ptolemy quoting Posidonius quoting Eratosthenes. But the remarkable Sanad ibn Ali had another trick up his sleeve. The great eleventh-century Muslim polymath al-Bīrūni reports in his famous treatise
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Determination of the Coordinates of Cities, al-Bīrūni begins his description of the method by referring back to the famous measurements carried out first by Eratosthenes and later repeated by al-Ma’mūn’s astronomers. Then, with his legendary sharp wit, he writes the immortal lines: ‘Here is another method for
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(c. 310–230 BCE), who stated correctly that the earth rotated around its own axis, and in turn revolved around the sun. Like his contemporary, Eratosthenes, Aristarchus had calculated the size of the earth, and estimated the size and distance of the moon and sun. From these, he concluded that the
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willingness to revise his own beliefs in the light of new evidence and his renowned unforgiving criticisms of sloppy reasoning of other scholars such as Eratosthenes place him, almost uniquely among the Greeks, as an early adherent of the scientific method. Indeed, he would not have looked out of place among
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the Spherical Earth’, Perspecta, 25 (1989), p. 3. 10. Thurston, ‘Greek Mathematical Astronomy Reconsidered’, p. 66. 11. Posidonius had carried out his own measurements following Eratosthenes’ method and agreed with his value of 250,000 stadia to begin with but then revised it downwards to 180,000, which is the value
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–8 Umayyads 25–6 see also chemistry; Jābir ibn Hayyān; transmutation Alexander the Great 45–6 Alexandria Bibliotheca Alexandrina 69–70 Library 69, 77–8; Eratosthenes 85 Alfraganus see al-Farghāni algebra 111–15 al-Khwārizmi 73, 77, 110–13 Diophantus 113, 115–17 Ibn al-Haytham 166 rhetorical 120, 121
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, fundamental 51–2, 64 Elephant Clock 228–9 emission theory 159–61 equants 210, 211, 216 equations Pell 117 quadratic 114–15, 120 quartic 166 Eratosthenes, size of the earth 85–6 ethics, medical 145, 250 Euclid Elements: translation 48, 81, 107, 117; use of quadratic equation 114, 115 emission theory
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bone 93 Leonardo of Pisa see Fibonacci, Leonardo libraries 69–70 Córdoba 192, 194 see also House of Wisdom Library of Alexandria 69, 77–8 Eratosthenes 85 Library of Nineveh 70 light, nature of 164 Macoraba 20 Maimonides, Moses 201 māl 119 al-Ma’mūn, Abū Ja’far Abdullah 4, 6
The Interstellar Age: Inside the Forty-Year Voyager Mission
by Jim Bell · 24 Feb 2015 · 310pp · 89,653 words
teach kids about timekeeping and understanding our place in space using only sticks and shadows—much like the third-century BCE Greek mathematician and astronomer Eratosthenes had done to accurately estimate the size of our planet. We figured, apparently as Carl Sagan did for the Voyager Golden Record, let’s keep
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Earth’s shadow appears curved. It seemed obvious to Pythagoras. It would take more than 250 years, however, for another famous Greek mathematician and astronomer, Eratosthenes, to prove it and to accurately estimate our planet’s size. He performed one of the most simple and famous scientific experiments of all time
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, the sun was directly overhead and that stick did not cast a shadow. The other stick was in his own northern Egyptian city of Alexandria (Eratosthenes was the head of the Library of Alexandria, an amazing collection of all of the then-known books of the world—the equivalent of the
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, not mathematical slouches, to be sure, used their best reasoning to estimate the diameter of the Earth as 14,000 and 11,000 miles, respectively. Eratosthenes, armed with data from his simple measurements, came up with around 9,000 miles, or within about 15 percent of the correct modern answer (7
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–239 Earthrise, 228–229 8-track tape player-recorder, 53–54 Elliot, Jim, 184 Enceladus, 131, 142, 146–147, 151, 158–159, 166, 241, 243 Eratosthenes, 225 Eris, 242, 243 Europa, 115, 120–125, 127, 130–132, 166, 241, 243 Extrasolar planets, 282–288 Extraterrestrial life, 75, 77, 80, 104–105
Prime Obsession:: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
by John Derbyshire · 14 Apr 2003
preparations for turning the Golden Key by offering an improved version of the PNT. 99 PRIME OBSESSION 100 II. It begins with the “sieve of Eratosthenes.” The Golden Key is, in fact, just a way that Leonhard Euler found to express the sieve of
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Eratosthenes in the language of analysis.34 Eratosthenes of Cyrene (nowadays the little town of Shahhat in Libya) was one of the librarians at the great library of Alexandria. Around 230 B.C.
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, and so on. If you keep doing this for ever, the numbers you are left with are all the primes. That is the sieve of Eratosthenes. If you stop just before processing prime p—that is, just before removing every pth number that wasn’t already removed—you have all the
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up to 72, which is 49. After that you see some numbers, like 77, that are not prime. PRIME OBSESSION 102 III. The sieve of Eratosthenes is pretty straightforward, and 2,230 years old. How does it get us into the middle of the nineteenth century, and deep results in function
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11 Note that writing it in this way involves writing out all the positive whole numbers—which is how we started off the sieve of Eratosthenes (except that this time I included 1). What I’m going to do is multiply both sides of the equals sign by 1 2s . This
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The subtraction eliminated all the even-numbered terms from the infinite sum. I’m left with just the odd-numbered terms. Remembering the sieve of Eratosthenes, I’ll now multiply both 1 sides of this equals sign by 3s , 3 being the first unscathed number on the right-hand side. 1
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multiples of 5 vanished in the subtraction, and the first number left unscathed on the right is 7. Notice the resemblance to the sieve of Eratosthenes? Actually, you should first notice the difference. When doing the original sieve, I chose to leave each original prime standing, deleting only its multiples by
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of the infinite sum run through all the positive whole numbers. I have showed how, by applying a process very much like the sieve of Eratosthenes to this sum, it is equivalent to ζ (s ) = 1 1 1− s 2 × 1 × 1 1− s 3 1 1 1− s 5 1
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Riemann’s 1859 paper, the one I developed in Chapter 7, arguing that it is just a fancy way to write out the sieve of Eratosthenes. 1 1 1 1 1 × × × × × ×K 1 1 1 1 1 1 1− s 1− s 1− s 1− s 1− s 1− s 13
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this further; it’s like tracing the genealogies of those German princes. Another Mendelssohn link will show up in Chapter 20.v. CHAPTER 7 34. “Eratosthenes” is pronounced—at any rate by mathematicians—“eraTOSS-the-niece.” NOTES 373 35. Mathematics allows infinite products, just as it allows infinite sums. As with
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-125, 198 Elizabeth, Empress of Russia (daughter of Peter the Great), 58 Encke, Johann Franz, 53-54 Entire functions, 332-333 ε, 74, 371 INDEX Eratosthenes of Cyrene, 100-101, 372 Erdős, Paul, 125, 378 Ernest Augustus, King of Hanover, 26, 366 Error term, 190, 234-235, 236-237, 241, 243
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, 72, 97, 135, 222 calculus version, 309-311 expression, 105, 138, 303-304 and Möbius function, 245-246 proof of, 102-104, 107 sieve of Eratosthenes and, 100-101 turning, 303-311 Gonek, Steve, xiv Gordan, Paul Albert, 185 Gordan’s Problem, 184 Göttingen, city of, 255-256, 383 Göttingen Seven
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, Jean-Pierre, 372, 384 Set theory, 18, 88 Seven Years War, 60 INDEX 420 Siegel, Carl, 256-257, 263-264, 383; pl. 5 Sieve of Eratosthenes, 100-101, 102104, 138 Sine function, 147, 332 Skewes, Samuel, 236 Skewes’ number, 236 Snaith, Nina, xiv Snowflake curve, 381 Society of German Scientists and
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, Hugh, 262 Wigner, Eugene, 282, 387 Wild Numbers, The (Schogt), 161 Wiles, Andrew, 90, 161, 245, 271, 354355 Y Yorke, James, 387 422 sieve of Eratosthenes and, 102-104, 138 values of, 79-81, 146-147, 263 visualization, 216-218 zeros of, 154, 160, 169, 190-192, 206, 211-212, 217
NumPy Cookbook
by Ivan Idris · 30 Sep 2012 · 197pp · 35,256 words
palindromic numbers The steady state vector determination Discovering a power law Trading periodically on dips Simulating trading at random Sieving integers with the Sieve of Eratosthenes Introduction This chapter is about the commonly used functions. These are the functions that you will be using on a daily basis. Obviously, the usage
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also The Installing Matplotlib recipe in Chapter 1, Winding Along with IPython Sieving integers with the Sieve of Erasthothenes The Sieve of Eratosthenes (http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes) is an algorithm that filters out prime numbers. It iteratively identifies multiples of found primes. This sieve is efficient for primes
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the arange function for that: a = numpy.arange(i, i + LIM, 2) Sieve out multiples of p.We are not sure if this is what Eratosthenes wanted us to do, but it works. In the following code, we are passing a NumPy array and getting rid of all the elements that
God Created the Integers: The Mathematical Breakthroughs That Changed History
by Stephen Hawking · 28 Mar 2007
Archimedes. Heiberg must have been amazed when he examined the palimpsest firsthand. Kerameus had found the long lost treatise, The Method, which begins, “Archimedes to Eratosthenes greeting.” The presence of other Archimedean works in the palimpsest only confirmed its authorship. The Kerameus–Heiberg palimpsest was originally written in the tenth century
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two angles, each less than a right angle, of which α is the greater, then THE METHOD OF ARCHIMEDES TREATING OF MECHANICAL PROBLEMS—TO ERATOSTHENES “Archimedes to Eratosthenes greeting. I sent you on a former occasion some of the theorems discovered by me, merely writing out the enunciations and inviting you to
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, 376–377 Archimedes, 106, 371, 1225 biography, 119–125 Measurement of the Circle, 121, 194–199 The Method of Archimedes Treating of Mechanical Problems - to Eratosthenes, 123–125, 209–239 The Sand Reckoner, 200–208 On the Sphere and Cylinder - Book I, 123, 126–167 On the Sphere and Cylinder - Book
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Perfect Method (Boole), 951–958 of exhaustion, 121 On a General Method in Analysis (Boole), 838 The Method of Archimedes Treating of Mechanical Problems - to Eratosthenes (Archimedes), 123–125, 209–239 middle terms, 913–914 minima of ordinates, 441 modulus associates, 617 composite number, 619 composition of several prime numbers, 631
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