To Explain the World: The Discovery of Modern Science (77 page)

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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.

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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.

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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.

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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.

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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.

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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.”

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His full name is Abū Abdallāh Muhammad ibn Mūsā al-Khwārizmī. Full Arab names tend to be long, so I will usually just give the abbreviated name by which these persons are generally known. I will also dispense with diacritical marks such as bars over vowels, as in ā, which have no significance for readers (like myself) ignorant of Arabic.

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Alfraganus is the latinized name by which al-Farghani became known in medieval Europe. In what follows, the latinized names of other Arabs will be given, as here, in parentheses.

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Al-Biruni actually used a mixed decimal and sexigesimal system of numbers. He gave the height of the mountain in cubits as 652;3;18, that is, 652 plus 3/60 plus 18/3,600, which equals 652.055 in modern decimal notation.

*
But see the
footnote
in
Chapter 10
.

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A later writer, Georg Hartmann (1489–1564), claimed that he had seen a letter by Regiomontanus containing the sentence “The motion of the stars must vary a tiny bit on account of the motion of the Earth” (
Dictonary of Scientific Biography
, Scribner, New York, 1975, Volume II, p. 351). If this is true, then Regiomontanus may have anticipated Copernicus, though the sentence is also consistent with the Pythagorean doctrine that the Earth and Sun both revolve around the center of the world.

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Butterfield coined the phrase “the Whig interpretation of history,” which he used to criticize historians who judge the past according to its contribution to our present enlightened practices. But when it came to the scientific revolution, Butterfield was thoroughly Whiggish, as am I.

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As mentioned in
Chapter 8
, there is only one special case of the simplest version of Ptolemy’s theory (with one epicycle for each planet, and none for the Sun) that is equivalent to the simplest version of the Copernican theory, differing only in point of view: it is the special case in which the deferents of the inner planets are each taken to coincide with the orbit of the Sun around the Earth, while the radii of the epicycles of the outer planets all equal the distance of the Sun from the Earth. The radii of the epicycles of the inner planets and the radii of the deferents of the outer planets in this special case of the Ptolemaic theory coincide with the radii of planetary orbits in the Copernican theory.

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There are 120 ways of choosing the order of any five different things; any of the five can be first, any of the remaining four can be second, any of the remaining three can be third, and any of the remaining two can be fourth, leaving only one possibility for the fifth, so the number of ways of arranging five things in order is 5 × 4 × 3 × 2 × 1 = 120. But as far as the ratio of circumscribed and inscribed spheres is concerned, the five regular polyhedrons are not all different; this ratio is the same for the cube and the octahedron, and for the icosahedron and the dodecahedron. Hence two arrangements of the five regular polyhedrons that differ only by the interchange of a cube and an octahedron, or of an icosahedron and a dodecahedron, give the same model of the solar system. The number of different models is therefore 120/(2 × 2) = 30.

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For instance, if a cube is inscribed within the inner radius of the sphere of Saturn, and circumscribed about the outer radius of the sphere of Jupiter, then the ratio of the minimum distance of Saturn from the Sun and the maximum distance of Jupiter from the Sun, which according to Copernicus was 1.586, should equal the distance from the center of a cube to any of its vertices divided by the distance from the center of the same cube to the center of any of its faces, or √3 = 1.732, which is 9 percent too large.

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The motion of Mars is the ideal test case for planetary theories. Unlike Mercury or Venus, Mars can be seen high in the night sky, where observations are easiest. In any given span of years, it makes many more revolutions in its orbit than Jupiter or Saturn. And its orbit departs from a circle more than that of any other major planet except Mercury (which is never seen far from the Sun and hence is difficult to observe), so departures from circular motion at constant speed are much more conspicuous for Mars than for other planets.

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The main effect of the ellipticity of planetary orbits is not so much the ellipticity itself as the fact that the Sun is at a focus rather than the center of the ellipse. To be precise, the distance between either focus and the center of an ellipse is proportional to the eccentricity, while the variation in the distance of points on the ellipse from either focus is proportional to the
square
of the eccentricity, which for a small eccentricity makes it much smaller. For instance, for an eccentricity of 0.1 (similar to that of the orbit of Mars) the smallest distance of the planet from the Sun is only ½ percent smaller than the largest distance. On the other hand, the distance of the Sun from the center of this orbit is 10 percent of the average radius of the orbit.

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This is Julius Caesar Scaliger, a passionate defender of Aristotle and opponent of Copernicus.

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A subsequent discussion shows that by the mean distance Kepler meant, not the distance averaged over time, but rather the average of the minimum and maximum distances of the planet from the Sun. As shown in
Technical Note 18
, the minimum and maximum distances of a planet from the Sun are (1 -
e
)
a
and (1 +
e
)
a
, where
e
is the eccentricity and
a
is half the longer axis of the ellipse (that is, the semimajor axis), so the mean distance is just
a.
It is further shown in
Technical Note 18
that this is also the distance of the planet from the Sun, averaged over the distance traveled by the planet in its orbit.

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Focal length is a length that characterizes the optical properties of a lens. For a convex lens, it is the distance behind the lens at which rays that enter the lens in parallel directions converge. For a concave lens that bends converging rays into parallel directions, the focal length is the distance behind the lens at which the rays would have converged if not for the lens. The focal length depends on the radius of curvature of the lens and on the ratio of the speeds of light in air and glass. (See
Technical Note 22
.)

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The angular size of planets is large enough so that the lines of sight from different points on a planetary disk are farther apart as they pass through the Earth’s atmosphere than the size of typical atmospheric fluctuations; as a result, the effects of the fluctuations on the light from different lines of sight are uncorrelated, and therefore tend to cancel rather than add coherently. This is why we do not see planets twinkle.

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It would have pained Galileo to know that these are the names that have survived to the present. They were given to the Jovian satellites in 1614 by Simon Mayr, a German astronomer who argued with Galileo over who had discovered the satellites first.

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Presumably Galileo was not using a clock, but rather observing the apparent motions of stars. Since the stars seem to go 360° around the Earth in a sidereal day of nearly 24 hours, a 1° change in the position of a star indicates a passage of time equal to
¹
/
₃₆₀
times 24 hours, or 4 minutes.

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This is actually true only for swings of the pendulum through small angles, though Galileo did not note this qualification. Indeed he speaks of swings of 50° or 60° (degrees of arc) taking the same time as much smaller swings, and this suggests that he did not actually do all the experiments on the pendulum that he reported.

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Taken literally, this would mean that a body dropped from rest would never fall, since with zero initial velocity at the end of the first infinitesimal instant it would not have moved, and hence with speed proportional to distance would still have zero velocity. Perhaps the doctrine that the speed is proportional to the distance fallen was intended to apply only after a brief initial period of acceleration.

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One of Galileo’s arguments is fallacious, because it applies to the
average
speed during an interval of time, not to the speed acquired by the end of that interval.

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This is shown in Technical Note 25. As explained there, though Galileo did not know it, the speed of the ball rolling down the plane is not equal to the speed of a body that would have fallen freely the same vertical difference, because some of the energy released by the vertical descent goes into the rotation of the ball. But the speeds are proportional, so Galileo’s qualitative conclusion that the speed of a falling body is proportional to the time elapsed is not changed when we take into account the ball’s rotation.

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Descartes compared light to a rigid stick, which when pushed at one end instantaneously moves at the other end. He was wrong about sticks too, though for reasons he could not then have known. When a stick is pushed at one end, nothing happens at the other end until a wave of compression (essentially a sound wave) has traveled from one end of the stick to the other. The speed of this wave increases with the rigidity of the stick, but Einstein’s special theory of relativity does not allow anything to be perfectly rigid; no wave can have a speed exceeding that of light. Descartes’ use of this sort of comparison is discussed by Peter Galison, “Descartes Comparisons: From the Invisible to the Visible,”
Isis
75, 311 (1984).