For biology, like geology but unlike chemistry, there is another problem. Living things are what they are not only because of the principles of physics, but also because of a vast number of historical accidents, including the accident that a comet or meteor hit the Earth 65 million years ago with enough impact to kill off the dinosaurs, and going back to the fact that the Earth formed at a certain distance from the Sun and with a certain initial chemical composition. We can understand some of these accidents statistically, but not individually. Kepler was wrong; no one will ever be able to calculate the distance of the Earth from the Sun solely from the principles of physics. What we mean by the unification of biology with the rest of science is only that there can be no freestanding principles of biology, any more than of geology. Any general principle of biology is what it is because of the fundamental principles of physics together with historical accidents, which by definition can never be explained.
The point of view described here is called (often disapprovingly) “reductionism.” There is opposition to reductionism even within physics. Physicists who study fluids or solids often cite examples of “emergence,” the appearance in the description of macroscopic phenomena of concepts like heat or phase transition that have no counterpart in elementary particle physics, and that do not depend on the details of elementary particles. For instance, thermodynamics, the science of heat, applies in a wide variety of systems: not just to those considered by Maxwell and Boltzmann, containing large numbers of molecules, but also to the surfaces of large black holes. But it does not apply to everything, and when we ask whether it applies to a given system and if so why, we must have reference to deeper, more truly fundamental, principles of physics. Reductionism in this sense is not a program for the reform of scientific practice; it is a view of why the world is the way it is.
We do not know how long science will continue on this reductive path. We may come to a point where further progress is impossible within the resources of our species. Right now, it seems that there is a scale of mass about a million trillion times larger than the mass of the hydrogen atom, at which gravity and other as yet undetected forces are unified with the forces of the Standard Model. (This is known as the “Planck mass”; it is the mass that particles would have to possess for their gravitational attraction to be as strong as the electrical repulsion between two electrons at the same separation.) Even if the economic resources of the human race were entirely at the disposal of physicists, we cannot now conceive of any way of creating particles with such huge masses in our laboratories.
We may instead run out of intellectual resources—humans may not be smart enough to understand the really fundamental laws of physics. Or we may encounter phenomena that in principle cannot be brought into a unified framework for all science. For instance, although we may well come to understand the processes in the brain responsible for consciousness, it is hard to see how we will ever describe conscious feelings themselves in physical terms.
Still, we have come a long way on this path, and are not yet at its end.6 This is a grand story—how celestial and terrestrial physics were unified by Newton, how a unified theory of electricity and magnetism was developed that turned out to explain light, how the quantum theory of electromagnetism was expanded to include the weak and strong nuclear forces, and how chemistry and even biology were brought into a unified though incomplete view of nature based on physics. It is toward a more fundamental physical theory that the wide-ranging scientific principles we discover have been, and are being, reduced.
Acknowledgments
I was fortunate to have the help of several learned scholars: the classicist Jim Hankinson and the historians Bruce Hunt and George Smith. They read through most of the book, and I made many corrections based on their suggestions. I am deeply grateful for this help. I am indebted also to Louise Weinberg for invaluable critical comments, and for suggesting the lines of John Donne that now grace this book’s front matter. Thanks, too, to Peter Dear, Owen Gingerich, Alberto Martinez, Sam Schweber, and Paul Woodruff for advice on specific topics. Finally, for encouragement and good advice, many thanks are due to my wise agent, Morton Janklow; and to my fine editors at HarperCollins, Tim Duggan and Emily Cunningham.
Technical Notes
The following notes describe the scientific and mathematical background for many of the historical developments discussed in this book. Readers who have learned some algebra and geometry in high school and have not entirely forgotten what they learned should have no trouble with the level of mathematics in these notes. But I have tried to organize this book so that readers who are not interested in technical details can skip these notes and still understand the main text.
A warning: The reasoning in these notes is not necessarily identical to that followed historically. From Thales to Newton, the style of the mathematics that was applied to physical problems was far more geometric and less algebraic than is common today. To analyze these problems in this geometric style would be difficult for me and tedious for the reader. In these notes I will show how the results obtained by the natural philosophers of the past do follow (or in some cases, do not follow) from the observations and assumptions on which they relied, but without attempting faithfully to reproduce the details of their reasoning.
Notes
1. Thales’ Theorem
2. Platonic Solids
3. Harmony
4. The Pythagorean Theorem
5. Irrational Numbers
6. Terminal Velocity
7. Falling Drops
8. Reflection
9. Floating and Submerged Bodies
10. Areas of Circles
11. Sizes and Distances of the Sun and Moon
12. The Size of the Earth
13. Epicycles for Inner and Outer Planets
14. Lunar Parallax
15. Sines and Chords
16. Horizons
17. Geometric Proof of the Mean Speed Theorem
18. Ellipses
19. Elongations and Orbits of the Inner Planets
20. Diurnal Parallax
21. The Equal-Area Rule, and the Equant
22. Focal Length
23. Telescopes
24. Mountains on the Moon
25. Gravitational Acceleration
26. Parabolic Trajectories
27. Tennis Ball Derivation of the Law of Refraction
28. Least-Time Derivation of the Law of Refraction
29. The Theory of the Rainbow
30. Wave Theory Derivation of the Law of Refraction
31. Measuring the Speed of Light
32. Centripetal Acceleration
33. Comparing the Moon with a Falling Body
34. Conservation of Momentum
35. Planetary Masses
1. Thales’ Theorem
Thales’ theorem uses simple geometric reasoning to derive a result about circles and triangles that is not immediately obvious. Whether or not Thales was the one who proved this result, it is useful to look at the theorem as an instance of the scope of Greek knowledge of geometry before the time of Euclid.
Consider any circle, and any diameter of it. Let A and B be the points where the diameter intersects the circle. Draw lines from A and B to any other point P on the circle. The diameter and the lines running from A to P and from B to P form a triangle, ABP. (We identify triangles by listing their three corner points.) Thales’ Theorem tells us that this is a right triangle: the angle of triangle ABP at P is a right angle, or in other terms, 90°.
The trick in proving this theorem is to draw a line from the center C of the circle to point P. This divides triangle ABP into two triangles, ACP and BCP. (See Figure 1.) Both of these are isosceles triangles, that is, triangles with two sides equal. In triangle ACP, sides CA and CP are both radii of the circle, which have the same length according to the definition of a circle. (We label the sides of a triangle by the corner points they connect.) Likewise, in triangle BCP, sides CB and CP are equal. In an isosceles t
riangle the angles adjoining the two equal sides are equal, so angle α (alpha) at the intersection of sides AP and AC is equal to the angle at the intersection of sides AP and CP, while angle β (beta) at the intersection of sides BP and BC is equal to the angle at the intersection of sides BP and CP. The sum of the angles of any triangle is two right angles,* or in familiar terms 180°, so if we take αʹ as the third angle of triangle ACP, the angle at the intersection of sides AC and CP, and likewise take βʹ as the angle at the intersection of sides BC and CP, then
2α + αʹ = 180° 2β + βʹ = 180°
Adding these two equations and regrouping terms gives
2(α + β) + (αʹ + βʹ) = 360°
Now, αʹ + βʹ is the angle between AC and BC, which come together in a straight line, and is therefore half a full turn, or 180°, so
2(α + β) = 360° − 180° = 180°
and therefore α + β = 90°. But a glance at Figure 1 shows that α + β is the angle between sides AP and BP of triangle ABP, with which we started, so we see that this is indeed a right triangle, as was to be proved.
Figure 1. Proof of Thales’ theorem. The theorem states that wherever point P is located on the circle, the angle between the lines from the ends of the diameter to P is a right angle.
2. Platonic Solids
In Plato’s speculations about the nature of matter, a central role was played by a class of solid shapes known as regular polyhedrons, which have come also to be known as Platonic solids. The regular polyhedrons can be regarded as three-dimensional generalizations of the regular polygons of plane geometry, and are in a sense built up from regular polygons. A regular polygon is a plane figure bounded by some number n of straight lines, all of which are of the same length and meet at each of the n corners with the same angles. Examples are the equilateral triangle (a triangle with all sides equal) and the square. A regular polyhedron is a solid figure bounded by regular polygons, all of which are identical, with the same number N of polygons meeting with the same angles at every vertex.
The most familiar example of a regular polyhedron is the cube. A cube is bounded by six equal squares, with three squares meeting at each of its eight vertices. There is an even simpler regular polyhedron, the tetrahedron, a triangular pyramid bounded by four equal equilateral triangles, with three triangles meeting at each of the four vertices. (We will be concerned here only with polyhedrons that are convex, with every vertex pointed outward, as in the case of the cube and the tetrahedron.) As we read in the Timaeus, it had somehow became known to Plato that these regular polyhedrons come in only five possible shapes, which he took to be the shapes of the atoms of which all matter is composed. They are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron, with 4, 6, 8, 12, and 20 faces, respectively.
The earliest attempt to prove that there are just five regular polyhedrons that has survived from antiquity is the climactic last paragraph of Euclid’s Elements. In Propositions 13 through 17 of Book XIII Euclid had given geometric constructions of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron. Then he states,* “I say next that no other figure, besides the said five figures, can be constructed which is contained by equilateral and equiangular figures equal to one another.” In fact, what Euclid actually demonstrates after this statement is a weaker result, that there are only five combinations of the number n of sides of each polygonal face, and the number N of polygons meeting at each vertex, that are possible for a regular polyhedron. The proof given below is essentially the same as Euclid’s, expressed in modern terms.
The first step is to calculate the interior angle θ (theta) at each of the n vertices of an n-sided regular polygon. Draw lines from the center of the polygon to the vertices on the boundary. This divides the interior of the polygon into n triangles. Since the sum of the angles of any triangle is 180°, and each of these triangles has two vertices with angles θ/2, the angle of the third vertex of each triangle (the one at the center of the polygon) must be 180° – θ. But these n angles must add up to 360°, so n (180° – θ) = 360°. The solution is
For instance, for an equilateral triangle we have n = 3, so θ = 180° – 120° = 60°, while for a square n = 4, so θ = 180° – 90° = 90°.
The next step is to imagine cutting all the edges and vertices of a regular polyhedron except at one vertex, and pushing the polyhedron down onto a plane at that vertex. The N polygons meeting at that vertex will then be lying in the plane, but there must be space left over or the N polygons would have formed a single face. So we must have Nθ < 360°. Using the above formula for θ and dividing both sides of the inequality by 360° then gives
or equivalently (dividing both sides by N),
Now, we must have n ≥ 3 because otherwise there would be no area between the sides of the polygons, and we must have N ≥ 3 because otherwise there would be no space between the faces coming together at a vertex. (For instance, for a cube n = 4 because the sides are squares, and N = 3.) Thus the above inequality does not allow either 1/n or 1/N to be as small as 1/2 – 1/3 = 1/6, and consequently neither n nor N can be as large as 6. We can easily check every pair of values of whole numbers 5 ≥ N ≥ 3 and 5 ≥ n ≥ 3 to see if they satisfy the inequality, and find that there are only five pairs that do:
(a) N = 3 , n = 3
(b) N = 4 , n = 3
(c) N = 5 , n = 3
(d) N = 3 , n = 4
(e) N = 3 , n = 5
(In the cases n = 3, n = 4, and n = 5 the sides of the regular polyhedron are respectively equilateral triangles, squares, and regular pentagons.) These are the values of N and n that we find in the tetrahedron, octahedron, icosahedron, cube, and dodecahedron.
This much was proved by Euclid. But Euclid did not prove that there is only one regular polyhedron for each pair of n and N. In what follows, we will go beyond Euclid, and show that for each value of N and n we can find unique results for the other properties of the polyhedron: the number F of faces, the number E of edges, and the number V of vertices. There are three unknowns here, so for this purpose we need three equations. To derive the first, note that the total number of borders of all the polygons on the surface of the polyhedron is nF, but each of the E edges borders two polygons, so
2E = nF
Also, there are N edges coming together at each of the V vertices, and each of the E edges connects two vertices, so
2E = NV
Finally, there is a more subtle relation among F, E, and V. In deriving this relation, we must make an additional assumption, that the polyhedron is simply connected, in the sense that any path between two points on the surface can be continuously deformed into any other path between these points. This is the case for instance for a cube or a tetrahedron, but not for a polyhedron (regular or not) constructed by drawing edges and faces on the surface of a doughnut. A deep theorem states that any simply connected polyhedron can be constructed by adding edges, faces, and/or vertices to a tetrahedron, and then if necessary continuously squeezing the resulting polyhedron into some desired shape. Using this fact, we shall now show that any simply connected polyhedron (regular or not) satisfies the relation:
F − E + V = 2
It is easy to check that this is satisfied for a tetrahedron, in which case we have F = 4, E = 6, and V = 4, so the left-hand side is 4 – 6 + 4 = 2. Now, if we add an edge to any polyhedron, running across a face from one edge to another, we add one new face and two new vertices, so F and V increase by one unit and two units, respectively. But this splits each old edge at the ends of the new edge into two pieces, so E increases by 1 + 2 = 3, and the quantity F – E + V is thereby unchanged. Likewise, if we add an edge that runs from a vertex to one of the old edges, then we increase F and V by one unit each, and E by two units, so the quantity F – E + V is still unchanged. Finally, if we add an edge that runs from one vertex to another vertex, then we increase both F and E by one unit each and do not change V, so again F – E + V is unchanged. Since any simply connected polyhedron can be built up in this
way, all such polyhedrons have the same value for this quantity, which therefore must be the same value F – E + V = 2 as for a tetrahedron. (This is a simple example of a branch of mathematics known as topology; the quantity F – E + V is known in topology as the “Euler characteristic” of the polyhedron.)
We can now solve these three equations for E, F, and V. It is simplest to use the first two equations to replace F and V in the third equation with 2E/n and 2E/N, respectively, so that the third equation becomes 2E/n – E + 2E/N = 2, which gives
Then using the other two equations, we have
Thus for the five cases listed above, the numbers of faces, vertices, and edges are:
These are the Platonic solids.
3. Harmony
The Pythagoreans discovered that two strings of a musical instrument, with the same tension, thickness, and composition, will make a pleasant sound when plucked at the same time, if the ratio of the strings’ lengths is a ratio of small whole numbers, such as 1/2, 2/3, 1/4, 3/4, etc. To see why this is so, we first need to work out the general relation between the frequency, wavelength, and velocity of any sort of wave.
Any wave is characterized by some sort of oscillating amplitude. The amplitude of a sound wave is the pressure in the air carrying the wave; the amplitude of an ocean wave is the height of the water; the amplitude of a light wave with a definite direction of polarization is the electric field in that direction; and the amplitude of a wave moving along the string of a musical instrument is the displacement of the string from its normal position, in a direction at right angles to the string.
There is a particularly simple kind of wave known as a sine wave. If we take a snapshot of a sine wave at any moment, we see that the amplitude vanishes at various points along the direction the wave is traveling. Concentrating for a moment on one such point, if we look farther along the direction of travel we will see that the amplitude rises and then falls again to zero, then as we look farther it falls to a negative value and rises again to zero, after which it repeats the whole cycle again and again as we look still farther along the wave’s direction. The distance between points at the beginning and end of any one complete cycle is a length characteristic of the wave, known as its wavelength, and conventionally denoted by the symbol λ (lambda). It will be important in what follows that, since the amplitude of the wave vanishes not only at the beginning and end of a cycle but also in the middle, the distance between successive vanishing points is half a wavelength, λ/2. Any two points where the amplitude vanishes therefore must be separated by some whole number of half wavelengths.
To Explain the World: The Discovery of Modern Science Page 28
