To Explain the World: The Discovery of Modern Science (78 page)

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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.

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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.

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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.

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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.

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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.

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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 3t
o
²
and
o
³
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.

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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.

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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).

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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.

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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.

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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.

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This may not have been known in the time of Thales, in which case the proof must be of a later date.

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This is from the standard translation by T. L. Heath (
Euclid’s Elements
, Green Lion Press, Santa Fe, N.M., 2002, p. 480).

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For a piano string there are small corrections due to the stiffness of the string; these corrections produce terms in
v
proportional to 1/
L
³
. I will ignore them here.

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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.

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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.