distances of Sun and Moon and, 296–301
lunar parallax and, 307
sines and chords and, 309–11
Tropic of Cancer, 76n, 302
Tusi couple, 117
Tycho Brahe, 40n, 107, 158–60, 165–67, 169, 176, 180, 182, 184–86, 188, 204, 251–52, 303–4, 307, 322–23
Tychonic system, 185–86, 251–52
Ummayad caliphate, 114, 116
uncertainty, 69, 197, 203–4, 239, 255
unification of forces, 34, 228–29, 260, 266–68
units of measure, 240–41
universe
expanding, 69, 83, 164, 249, 265
solar system seen as, 164
universities, 127, 131
Uraniborg observatory, 159–61
uranium, 11
Uranus, 250
Urban VII, Pope, 184–86
Ursa Major, 56–57
Ursa Minor, 57
vacuum (void), 25–26, 131, 134, 134, 180, 197–200, 204
van Helden, A., 372, 378, 381
Vatican, 128, 179, 183
velocity
acceleration and, 138–39
distance and, 223, 224n
momentum and, 232
Venus, 77, 245n
apparent brightness, 87, 142–43
apparent retrograde motion of, 148
Aristotle on, 84–85
Copernicus and, 86, 148–51, 155
distance from Earth, 142–43
eccentricity of orbit, 167
elongations and orbit of, 320–21
epicycles and deferents and, 303–5
Galileo and, 179–80, 204
Greeks and, 81–82, 84–90, 92, 94, 124
Kepler and, 162, 171
phases of, 142–43, 179–80, 204
Ptolemy and, 88–90, 92, 94, 155, 255
sidereal period, 171
transit across Sun, 240
Viète, François, 206
Vitruvius, 35
Vlastos, Gregory, 6n
Volta, Alessandro, 257
Voltaire, 204, 215n, 245, 248
Wallace, Alfred Russel, 24, 265–66
water
acceleration of falling, 33
bodies submerged in, 35, 38, 291–94
chemical composition of, 259
as element, 4, 6, 10, 12, 64–66, 259
falling, 33, 288–89
Plato on, 10, 12
Watson, Richard, 213, 381
wave, amplitude and velocity of, 279–81
weak forces, 257, 262
Weinberg, L., 269
Weinberg, S., 368, 371, 372, 373, 377, 379, 382
Westfall, R. S., 242, 376, 377, 381, 382
White, Andrew Dickson, 155, 376, 377
Whiteside, D. T., 381
Wigner, Eugene, 20, 368
William IV, landgrave of Hesse-Cassel, 159
William of Moerbeke, 129
Witten, Edward, 20
Woodruff, P., 269
Works and Days (Hesiod), 55
Wren, Christopher, 217, 231
Xenophanes, 6, 12, 14, 46, 65
year, length of, 59–60, 107
Young, Thomas, 222
Zeno, 8, 11, 23
zero, 126
zodiac, 57–58, 77, 81, 114
About the Author
Photo by Matt Valentine
STEVEN WEINBERG is a theoretical physicist and winner of the Nobel Prize in Physics, the National Medal of Science, the Lewis Thomas Prize for the Scientist as Poet, and numerous honorary degrees and other awards. He is a member of the National Academy of Science, the Royal Society of London, the American Philosophical Society, and other academies. A longtime contributor to The New York Review of Books, he is the author of The First Three Minutes, Dreams of a Final Theory, Facing Up, and Lake Views, as well as leading treatises in theoretical physics. He holds the Josey Regental Chair in Science at the University of Texas at Austin.
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Also by Steven Weinberg
Lectures on Quantum Mechanics
Lake Views
Cosmology
Glory and Terror
Facing Up
The Quantum Theory of Fields
Dreams of a Final Theory
Elementary Particles and the Laws of Physics
The Discovery of Subatomic Particles
The First Three Minutes
Gravitation and Cosmology
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* As pointed out by Gregory Vlastos, in Plato’s Universe (University of Washington Press, Seattle, 1975), an adverbial form of the word kosmos was used by Homer to mean “socially decent” and “morally proper.” This use survives in English in the word “cosmetic.” Its use by Heraclitus reflects the Hellenic view that the world is pretty much what it should be. The word appears in English also in the cognates “cosmos” and “cosmology.”
* In fact (as discussed in Technical Note 2), whatever may have been proved by Theaetetus, Elements does not prove what it claims to prove, that there are only five possible convex regular solids. Elements does prove that for regular polyhedrons, there are just five combinations of the number of sides of each face of a polyhedron and of the number of faces that meet at each vertex, but it does not prove that for each combination of these numbers there is just one possible convex regular polyhedron.
* The Greek word kineson, which is usually translated as “motion,” actually has a more general significance, referring to any sort of change. Thus Aristotle’s classification of types of cause applied not only to change of position, but to any change. The Greek word fora refers specifically to change of location, and is usually translated as “locomotion.”
* It was generally supposed in the ancient world that when we see something the light travels from the eye to the object, as if vision were a sort of touching that requires us to reac
h out to what is seen. In the following discussion I will take for granted the modern understanding, that in vision light travels from the object to the eye. Fortunately, in analyzing reflection and refraction, it makes no difference which way the light is going.
* The Assayer is a polemic against Galileo’s Jesuit adversaries, taking the form of a letter to the papal chamberlain Virginio Cesarini. As we will see in Chapter 11, Galileo in The Assayer was attacking the correct view of Tycho Brahe and the Jesuits that comets are farther from Earth than the Moon is. (The quotation here is taken from the translation by Maurice A. Finocchiaro, in The Essential Galileo, Hackett, Indianapolis, Ind., 2008, p. 183.)
* Pierre Gassendi was a French priest and philosopher who tried to reconcile the atomism of Epicurus and Lucretius with Christianity.
* To be more precise, this is known as the “synodic” lunar month. The 27-day period for the Moon to return to the same position relative to the fixed stars is known as the “sidereal” lunar month.
* This does not happen every month, because the plane of the orbit of the Moon around the Earth is slightly tilted with respect to the plane of the orbit of the Earth around the Sun. The Moon crosses the plane of the Earth’s orbit twice every sidereal month, but this happens at full moon, when the Earth is between the Sun and the Moon, only about once every 18 years.
* The equinox is the moment when the Sun in its motion against the background of stars crosses the celestial equator. (In modern terms, it is the moment when the line between the Earth and the Sun becomes perpendicular to the Earth’s axis.) At points on the Earth with different longitude, this moment occurs at different times of day, so there may be a one-day difference in the date that different observers report the equinox. Similar remarks apply to the phases of the Moon.
* It has been argued (in O. Neugebauer, A History of Ancient Mathematical Astronomy, Springer-Verlag, New York, 1975, pp. 1093–94), that Aristotle’s reasoning about the shape of the Earth’s shadow on the Moon is inconclusive, since an infinite variety of terrestrial and lunar shapes would give the same curved shadow.
* Samuel Eliot Morison cited this argument in his biography of Columbus (Admiral of the Ocean Sea, Little Brown, Boston, Mass., 1942) to show, contrary to a widespread supposition, that it was well understood before Columbus set sail that the Earth is a sphere. The debate in the court of Castile over whether to support the proposed expedition of Columbus concerned not the shape of the Earth, but its size. Columbus thought the Earth was small enough so that he could sail from Spain to the east coast of Asia without running out of food and water. He was wrong about the size of the Earth, but of course was saved by the unexpected appearance of America between Europe and Asia.
* There is a fascinating remark by Archimedes in The Sand Reckoner, that Aristarchus had found that the “Sun appeared to be about 1/720 part of the zodiac” (The Works of Archimedes, trans. T. L. Heath, Cambridge University Press, Cambridge, 1897, p. 223). That is, the angle subtended on Earth by the disk of the Sun is 1/720 times 360°, or 0.5°, not far from the correct value 0.519°. Archimedes even claimed that he had verified this by his own observations. But as we have seen, in his surviving work Aristarchus had given the angle subtended by the disk of the Moon the value 2°, and he had noted that the disks of the Sun and Moon have the same apparent size. Was Archimedes quoting a later measurement by Aristarchus, of which no report has survived? Was he quoting his own measurement, and attributing it to Aristarchus? I have heard scholars suggest that the source of the discrepancy is a copying error or a misinterpretation of the text, but this seems very unlikely. As already noted, Aristarchus had concluded from his measurement of the angular size of the Moon that its distance from the Earth must be between 30 and 45/2 times greater than the Moon’s diameter, a result quite incompatible with an apparent size of around 0.5°. Modern trigonometry tells us on the other hand that if the Moon’s apparent size were 2°, then its distance from the Earth would be 28.6 times its diameter, a number that is indeed between 30 and 45/2. (The Sand Reckoner is not a serious work of astronomy, but a demonstration by Archimedes that he could calculate very large numbers, such as the number of grains of sand needed to fill the sphere of the fixed stars.)
* There is a famous ancient device known as the Antikythera Mechanism, discovered in 1901 by sponge divers off the island of Antikythera, in the Mediterranean between Crete and mainland Greece. It is believed to have been lost in a shipwreck sometime around 150 to 100 BC. Though the Antikythera Mechanism is now a corroded mass of bronze, scholars have been able to deduce its workings by X-ray studies of its interior. Apparently it is not an orrery but a calendrical device, which tells the apparent position of the Sun and planets in the zodiac on any date. The most important thing about it is that its intricate gearwork provides evidence of the high competence of Hellenistic technology.
* The celestial latitude is the angular separation between the star and the ecliptic. While on Earth we measure longitude from the Greenwich meridian, the celestial longitude is the angular separation, on a circle of fixed celestial latitude, between the star and the celestial meridian on which lies the position of the Sun at the 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 the 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 tropical circle by Cleomedes), the circle on the Earth’s surface where the noon Sun at the summer solstice really is directly overhead. Alexandria is at a latitude of 31.2°, while the latitude of the Tropic of Cancer is 23.5°, which is less than the latitude of Alexandria by 7.7°, so the circumference of the Earth is in fact 360°/7.7° = 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.
* We can see the correspondence of days of the week with planets and the associated gods in the names of the days of the week in English. Saturday, Sunday, and Monday are obviously associated with Saturn, the Sun, and the Moon; Tuesday, Wednesday, Thursday, and Friday are based on an association of Germanic gods with supposed Latin equivalents: Tyr with Mars, Wotan with Mercury, Thor with Jupiter, and Frigga with Venus.
* In a year of 365¼ days, the Earth actually rotates on its axis 366¼ times. The Sun seems to go around the Earth only 365¼ times in this period, because at the same time that the Earth is rotating 366¼ times on its axis, it is going around the Sun once in the same direction, giving 365¼ apparent revolutions of the Sun around the Earth. Since it takes 365.25 days of 24 hours for the Earth to spin 366.25 times relative to the stars, the time it takes the Earth to spin once is (365.25 × 24 hours)/366.25, or 23 hours, 56 minutes, and 4 seconds. This is known as the sidereal day.
* The apparent luminosity of stars in catalogs from Ptolemy’s time to the present is described in terms of their “magnitude.” Magnitude increases with decreasing luminosity. The brightest star, Sirius, has magnitude -1.4, the bright star Vega has magnitude zero, and stars that are just barely visible to the naked eye are of sixth magnitude. In 1856 the astronomer Norman Pogson compared the measured apparent luminosity of a number of stars with the magnitudes that had historically been attributed to them, and on that basis decreed that if one star has a magnitude greater than another by 5 units, it is 100 times dimmer.
* In one of the few hints to the origin of the use of epicycles, Ptolemy at the beginning of Book XII of the Almagest credits Apollonius of Perga with proving a theorem relating the use of epicycles and
eccentrics in accounting for the apparent motion of the Sun.
* In the theory of the Sun’s motion an eccentric can be regarded as a sort of epicycle, on which the line from the center of the epicycle to the Sun is always parallel to the line between the Earth and the center of the Sun’s deferent, thus shifting the center of the Sun’s orbit away from the Earth. Similar remarks apply to the Moon and planets.
* The term “equant” was not used by Ptolemy. He referred instead to a “bisected eccentric,” indicating that the center of the deferent is taken to be in the middle of the line connecting the equant and the Earth.
* The same is true when eccentrics and equants are added; observation could have fixed only the ratios of the distances of the Earth and the equant from the center of the deferent and the radii of the deferent and epicycle, separately for each planet.
* The association of astrology with the Babylonians is illustrated in Ode XI of Book 1 of Horace: “Do not inquire (we are not allowed to know) what ends the gods have assigned to you and me, Leoconoe, and do not meddle with Babylonian horoscopes. How much better to endure whatever it proves to be.” (Horace, Odes and Epodes, ed. and trans. Niall Rudd, Loeb Classical Library, Harvard University Press, Cambridge, Mass., 2004, pp. 44–45). It sounds better in Latin: “Tu ne quaesieris—scire nefas—quem mihi, quem tibi, finem di dederint, Leuconoë, nec Babylonios temptaris numerous, ut melius, quidquid erit, pati.”
To Explain the World: The Discovery of Modern Science Page 42
