* 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.
* 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.
* 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.
* 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.
* 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.
* 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.
* 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.
* 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.
* 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.
* 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.
* This is Julius Caesar Scaliger, a passionate defender of Aristotle and opponent of Copernicus.
* 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.
* 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.)
* 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.
* 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.
* 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.
* 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.
* 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.
* 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.
* 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 el
apsed is not changed when we take into account the ball’s rotation.
* 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).
* Recall that the sine of an angle is the side opposite that angle in a right triangle, divided by the hypotenuse of the triangle. It increases as the angle increases from zero to 90°, in proportion to the angle for small angles, and then more slowly.
* This is done by finding the value of b/R where an infinitesimal change in b produces no change in φ, so that at that value of φ the graph of φ versus b/R is flat. This is the value of b/R where φ reaches its maximum value. (Any smooth curve like the graph of φ against b/R that rises to a maximum and then falls again must be flat at the maximum. A point where the curve is not flat cannot be the maximum, since if the curve at some point rises to the right or left there will be points to the right or left where the curve is higher.) Values of φ in the range where the curve of φ versus b/R is nearly flat vary only slowly as we vary b/R, so there are relatively many rays with values of φ in this range.
* In his fifties, Newton hired his half sister’s beautiful daughter, Catherine Barton, as his housekeeper, but though they were close friends they do not seem to have been romantically attached. Voltaire, who was in England at the time of Newton’s death, reported that Newton’s doctor and “the surgeon in whose arms he died” confirmed to Voltaire that Newton never had intimacies with a woman (see Voltaire, Philosophical Letters, Bobbs-Merrill Educational Publishing, Indianapolis, Ind., 1961, p. 63). Voltaire did not say how the doctor and surgeon could have known this.
* This is from a speech, “Newton, the Man,” that Keynes was to give at a meeting at the Royal Society in 1946. Keynes died three months before the meeting, and the speech was given by his brother.
* Newton devoted a comparable effort to experiments in alchemy. This could just as well be called chemistry, as between the two there was then no meaningful distinction. As remarked in connection with Jabir ibn Hayyan in Chapter 9, until the late eighteenth century there was no established chemical theory that would rule out the aims of alchemy, like the transmutation of base metals into gold. Although Newton’s work on alchemy thus did not represent an abandonment of science, it led to nothing important.
* A flat piece of glass does not separate the colors, because although each color is bent by a slightly different angle on entering the glass, they are all bent back to their original direction on leaving it. Because the sides of a prism are not parallel, light rays of different color that are refracted differently on entering the glass reach the prism’s surface on leaving the prism at angles that are not equal to the angles of refraction on entering, so when these rays are bent back on leaving the prism the different colors are still separated by small angles.
* This is the “natural logarithm” of 1 + x, the power to which the constant e = 2.71828 . . . must be raised to give the result 1 + x. The reason for this peculiar definition is that the natural logarithm has some properties that are much simpler than those of the “common logarithm,” in which 10 takes the place of e. For instance, Newton’s formula shows that the natural logarithm of 2 is given by the series 1 - ½ + ⅓ - ¼ + . . . , while the formula for the common logarithm of 2 is more complicated.
* The neglect of the terms 3to2 and o3 in this calculation may make it seem that the calculation is only approximate, but that is misleading. In the nineteenth century mathematicians learned to dispense with the rather vague idea of an infinitesimal o, and to speak instead of precisely defined limits: the velocity is the number to which [D(t + o) - D(t)]/o can be made as close as we like by taking o sufficiently small. As we will see, Newton later moved away from infinitesimals and toward the modern idea of limits.
* Kepler’s three laws of planetary motion were not well established before Newton, though the first law—that each planetary orbit is an ellipse with the Sun at one focus—was widely accepted. It was Newton’s derivation of these laws in the Principia that led to the general acceptance of all three laws.
* The first reasonably precise measurement of the circumference of the Earth was made around 1669 by Jean-Félix Picard (1620–1682), and was used by Newton in 1684 to improve this calculation.
* Newton was unable to solve the three-body problem of the Earth, Sun, and Moon with enough accuracy to calculate the peculiarities in the motion of the Moon that had worried Ptolemy, Ibn al-Shatir, and Copernicus. This was finally accomplished in 1752 by Alexis-Claude Clairaut, who used Newton’s theories of motion and gravitation.
* In Book III of the Opticks, Newton expressed the view that the solar system is unstable, and requires occasional readjustment. The question of the stability of the solar system remained controversial for centuries. In the late 1980s Jacques Laskar showed that the solar system is chaotic; it is impossible to predict the motions of Mercury, Venus, Earth, and Mars for more than about 5 million years into the future. Some initial conditions lead to planets colliding or being ejected from the solar system after a few billion years, while others that are nearly indistinguishable do not. For a review, see J. Laskar, “Is the Solar System Stable?,” www.arxiv.org/1209.5996 (2012).
* Maxwell himself did not write equations governing electric and magnetic fields in the form known today as “Maxwell’s equations.” His equations instead involved other fields known as potentials, whose rates of change with time and position are the electric and magnetic fields. The more familiar modern form of Maxwell’s equations was given around 1881 by Oliver Heaviside.
* Here and in what follows I will not cite individual physicists. So many are involved that it would take too much space, and many are still alive, so that I would risk giving offense by citing some physicists and not others.
* I am here lumping sexual selection together with natural selection, and punctuated equilibrium along with steady evolution; and I am not distinguishing between mutations and genetic drift as a source of inheritable variations. These distinctions are very important to biologists, but they do not affect the point that concerns me here: there is no independent law of biology that makes inheritable variations more likely to be improvements.
* This may not have been known in the time of Thales, in which case the proof must be of a later date.
* This is from the standard translation by T. L. Heath (Euclid’s Elements, Green Lion Press, Santa Fe, N.M., 2002, p. 480).
* For a piano string there are small corrections due to the stiffness of the string; these corrections produce terms in v proportional to 1/L3. I will ignore them here.
* In some musical scales middle G is given a slightly different frequency in order to make possible other pleasant chords involving middle G. The adjustment of frequencies to make as many chords as possible pleasant is called “tempering” the scale.
* This table appears in the translation of the Almagest by G. J. Toomer (Ptolemy’s Almagest, Duckworth, London, 1984, pp. 57–60).
* Galileo used a definition of “mile” that is not very different from the modern English mile. In modern units, the radius of the Moon is actually 1,080 miles.
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To Explain the World: The Discovery of Modern Science Page 43
