Now, let us consider the outer planets, Mars, Jupiter, and Saturn. In the simplest version of the theory of Copernicus (or Tycho) each planet keeps a fixed distance not only from the Sun, but also from a moving point Cʹ in space, which keeps a fixed distance from the Earth. To find this point, draw a parallelogram (Figure 7b), whose first three vertices in order around it are the position S of the Sun, the position E of the Earth, and the position Pʹ of one of the planets. The moving point Cʹ is the empty fourth corner of the parallelogram. Since the line between E and S has a fixed length, and the line between Pʹ and Cʹ is the opposite side of the parallelogram, it has an equal fixed length, so the planet stays at a fixed distance from Cʹ, equal to the distance of the Earth from the Sun. Likewise, since the line between S and pʹ has a fixed length, and the line between E and Cʹ is the opposite side of the parallelogram, it has an equal fixed length, so point Cʹ stays at a fixed distance from the Earth, equal to the distance of the planet from the Sun. This is a special case of the theory of Ptolemy, though a case never considered by him, in which the deferent is nothing but the orbit of point Cʹ around the Earth, and the epicycle is the orbit of Mars, Jupiter, or Saturn around Cʹ.
Figure 7. A simple version of the epicycle theory described by Ptolemy. (a) The supposed motion of one of the inner planets, Mercury or Venus. (b) The supposed motion of one of the outer planets Mars, Jupiter, or Saturn. Planet P goes on an epicycle around point C in one year, with the line from C to P always parallel to the line from the Earth to the Sun, while point C goes around the Earth on the deferent in a longer time. (The dashed lines indicate a special case of the Ptolemaic theory, for which it is equivalent to that of Copernicus.)
Once again, as far as the apparent position in the sky of the Sun and planets is concerned, we can multiply the changing distance of any planet from the Earth by a constant without changing appearances, by multiplying the radii of both the epicycle and the deferent by a constant factor, chosen independently for each outer planet. Although we no longer have a parallelogram, the line between the planet and C remains parallel to the line between the Earth and Sun. The apparent motion of each outer planet in the sky will be unchanged by this transformation, as long as we don’t change the ratio of the radii of each planet’s deferent and epicycle. This is a simple version of the theory proposed by Ptolemy for the outer planets. According to this theory, the planet goes around C on its epicycle in 1 year, while C goes around the deferent in the time that it actually takes the planet to go around the Sun: 1.9 years for Mars, 12 years for Jupiter, and 29 years for Saturn.
Specifically, since we do not change the ratio of the radii of the deferent and epicycle, we must now have
rEPI/rDEF = rE/rP
where rEPI and rDEF are again the radii of the epicycle and deferent in Ptolemy’s scheme, and rP and rE are the radii of the orbits of the planet and the Earth in the theory of Copernicus (or equivalently, the radii of the orbits of the planet around the Sun and the Sun around the Earth in the theory of Tycho). Once again, the above discussion describes, not how Ptolemy obtained his theory, but only why this theory worked so well.
14. Lunar Parallax
Suppose that the direction to the Moon is observed from point O on the surface of the Earth to be at angle ζʹ (zeta prime) to the zenith at O. The Moon moves in a smooth and regular way around the center of the Earth, so by using the results of repeated observations of the Moon it is possible to calculate the direction from the center C of the Earth to the Moon M at the same moment, and in particular to calculate angle ζ between the direction from C to the Moon and the direction of the zenith at O, which is the same as the direction of the line from the center of the Earth to O. Angles ζ and ζʹ differ slightly because the radius re of the earth is not entirely negligible compared with the distance d of the Moon from the center of the Earth, and from this difference Ptolemy could calculate the ratio d/re.
Points C, O, and M form a triangle, in which the angle at C is ζ, the angle at O is 180° – ζʹ, and (since the sum of the angles of any triangle is 180°) the angle at M is 180° – ζ – (180° – ζʹ) = ζʹ – ζ. (See Figure 8.) We can calculate the ratio d/re from these angles much more easily than Ptolemy did, by using a theorem of modern trigonometry: that in any triangle the lengths of sides are proportional to the sines of the opposite angles. (Sines are discussed in Technical Note 15.) The angle opposite the line of length re from C to O is ζʹ – ζ, and the angle opposite the line of length d from C to M is 180° – ζʹ, so
Figure 8. Use of parallax to measure the distance to the Moon. Here ζ’ is the observed angle between the line of sight to the Moon and the vertical direction, and ζ is the value this angle would have if the Moon were observed from the center of the Earth.
On October 1, AD 135, Ptolemy observed that the zenith angle of the Moon as seen from Alexandria was ζʹ = 50°55’, and his calculations showed that at the same moment the corresponding angle that would be observed from the center of the Earth was ζ = 49°48’. The relevant sines are
sinζʹ = 0.776 sin(ζʹ − ζ) = 0.0195
From this Ptolemy could conclude that the distance from the center of the Earth to the Moon in units of the radius of the Earth is
This was considerably less than the actual ratio, which on average is about 60. The trouble was that the difference ζʹ – ζ was not actually known accurately by Ptolemy, but at least it gave a good idea of the order of magnitude of the distance to the Moon.
Anyway, Ptolemy did better than Aristarchus, from whose values for the ratio of the diameters of the Earth and Moon and of the distance and diameter of the Moon he could have inferred that d/re is between 215/9 = 23.9 and 57/4 = 14.3. But if Aristarchus had used a correct value of about 1/2° for the angular diameter of the Moon’s disk instead of his value of 2° he would have found d/re to be four times greater, between 57.2 and 95.6. That range includes the true value.
15. Sines and Chords
The mathematicians and astronomers of antiquity could have made great use of a branch of mathematics known as trigonometry, which is taught today in high schools. Given any angle of a right triangle (other than the right angle itself) trigonometry tells us how to calculate the ratios of the lengths of all the sides. In particular, the side opposite the angle divided by the hypotenuse is a quantity known as the “sine” of that angle, which can be found by looking it up in mathematical tables or typing the angle in a hand calculator and pressing “sin.” (The side of the triangle adjacent to an angle divided by the hypotenuse is the “cosine” of the angle, and the side opposite divided by the side adjacent is the “tangent” of the angle, but it will be enough for us to deal here with sines.) Though no notion of a sine appears anywhere in Hellenistic mathematics, Ptolemy’s Almagest makes use of a related quantity, known as the “chord” of an angle.
To define the chord of an angle θ (theta), draw a circle of radius 1 (in whatever units of length you find convenient), and draw two radial lines from the center to the circumference, separated by that angle. The chord of the angle is the length of the straight line, or chord, that connects the points where the two radial lines intersect the circumference. (See Figure 9.) The Almagest gives a table of chords* in a Babylonian sexigesimal notation, with angles expressed in degrees of arc, running from 1/2° to 180°. For instance, the chord of 45° is given as 45 15 19, or in modern notation
while the true value is 0.7653669. . . .
The chord has a natural application to astronomy. If we imagine the stars as lying on a sphere of radius equal to 1, centered on the center of the Earth, then if the lines of sight to two stars are separated by angle θ, the apparent straight-line distance between the stars will be the chord of θ.
To see what these chords have to do with trigonometry, return to the figure used to define the chord of angle θ, and draw a line (the dashed line in Figure 9) from the center of the circle that just bisects the chord. We then have two right triangles, each with an angle at the center of
the circle equal to θ/2, and a side opposite this angle whose length is half the chord. The hypotenuse of each of these triangles is the radius of the circle, which we are taking to be 1, so the sine of θ/2, in mathematical notation sin(θ/2), is half the chord of θ, or:
chord of θ = 2sin(θ/2)
Hence any calculation that can be done with sines can also be done with chords, though in most cases less conveniently.
Figure 9. The chord of an angle θ. The circle here has a radius equal to 1. The solid radial lines make an angle θ at the center of the circle; the horizontal line runs between the intersections of these lines with the circle; and its length is the chord of this angle.
16. Horizons
Normally our vision outdoors is obstructed by nearby trees or houses or other obstacles. From the top of a hill, on a clear day, we can see much farther, but the range of our vision is still restricted by a horizon, beyond which lines of sight are blocked by the Earth itself. The Arab astronomer al-Biruni described a clever method for using this familiar phenomenon to measure the radius of the Earth, without needing to know any distances other than the height of the mountain.
Figure 10. Al-Biruni’s use of horizons to measure the size of the Earth. O is an observer on a hill of height h; H is the horizon as seen by this observer; the line from H to O is tangent to the Earth’s surface at H, and therefore makes a right angle with the line from the center C of the Earth to H.
An observer O on top of a hill can see out to a point H on the Earth’s surface, where the line of sight is tangent to the surface. (See Figure 10.) This line of sight is at right angles to the line joining H to the Earth’s center C, so triangle OCH is a right triangle. The line of sight is not in the horizontal direction, but below the horizontal direction by some angle θ, which is small because the Earth is large and the horizon far away. The angle between the line of sight and the vertical direction at the hill is then 90° – θ, so since the sum of the angles of any triangle must be 180°, the acute angle of the triangle at the center of the Earth is 180° – 90° – (90° – θ) = θ. The side of the triangle adjacent to this angle is the line from C to H, whose length is the Earth’s radius r, while the hypotenuse of the triangle is the distance from C to O, which is r + h, where h is the height of the mountain. According to the general definition of the cosine, the cosine of any angle is the ratio of the adjacent side to the hypotenuse, which here gives
To solve this equation for r, note that its reciprocal gives 1 + h/r = 1/cosθ, so by subtracting 1 from this equation and then taking the reciprocal again we have
For instance, on a mountain in India al-Biruni found θ = 34’, for which cosθ = 0.999951092 and 1/cosθ – 1 = 0.0000489. Hence
Al-Biruni reported that the height of this mountain is 652.055 cubits (a precision much greater than he could possibly have achieved), which then actually gives r = 13.3 million cubits, while his reported result was 12.8 million cubits. I don’t know the source of al-Biruni’s error.
17. Geometric Proof of the Mean Speed Theorem
Suppose we make a graph of speed versus time during uniform acceleration, with speed on the vertical axis and time on the horizontal axis. The graph will be a straight line, rising from zero speed at zero time to the final speed at the final time. In each tiny interval of time, the distance traveled is the product of the speed at that time (this speed changes by a negligible amount during that interval if the interval is short enough) times the time interval. That is, the distance traveled is equal to the area of a thin rectangle, whose height is the height of the graph at that time and whose width is the tiny time interval. (See Figure 11a.) We can fill up the area under the graph, from the initial to the final time, by such thin rectangles, and the total distance traveled will then be the total area of all these rectangles—that is, the area under the graph. (See Figure 11b.)
Figure 11. Geometric proof of the mean speed theorem. The slanted line is the graph of speed versus time for a body uniformly accelerated from rest. (a) The width of the small rectangle is a short time interval; its area is close to the distance traveled in that interval. (b) Time during a period of uniform acceleration, broken into short intervals; as the number of rectangles is increased the sum of the areas of the rectangles becomes arbitrarily close to the area under the slanted line. (c) The area under the slanted line is half the product of the elapsed time and the final speed.
Of course, however thin we make the rectangles, it is only an approximation to say that the area under the graph equals the total area of the rectangles. But we can make the rectangles as thin as we like, and in this way make the approximation as good as we like. By imagining the limit of an infinite number of infinitely thin rectangles, we can conclude that the distance traveled equals the area under the graph of speed versus time.
So far, this argument would be unchanged if the acceleration was not uniform, in which case the graph would not be a straight line. In fact, we have just deduced a fundamental principle of integral calculus: that if we make a graph of the rate of change of any quantity versus time, then the change in this quantity in any time interval is the area under the curve. But for a uniformly increasing rate of change, as in uniform acceleration, this area is given by a simple geometric theorem.
The theorem says that the area of a right triangle is half the product of the two sides adjacent to the right angle—that is, the two sides other than the hypotenuse. This follows immediately from the fact that we can put two of these triangles together to form a rectangle, whose area is the product of its two sides. (See Figure 11c.) In our case, the two sides adjacent to the right angle are the final speed and the total time elapsed. The distance traveled is the area of a right triangle with those dimensions, or half the product of the final speed and the total time elapsed. But since the speed is increasing from zero at a constant rate, its mean value is half its final value, so the distance traveled is the mean speed multiplied by the time elapsed. This is the mean speed theorem.
18. Ellipses
An ellipse is a certain kind of closed curve on a flat surface. There are at least three different ways of giving a precise description of this curve.
Figure 12. The elements of an ellipse. The marked points within the ellipse are its two foci; a and b are half the longer and shorter axes of the ellipse; and the distance from each focus to the center of the ellipse is ea. The sum of the lengths r+ and r– of the two lines from the foci to a point P equals 2a wherever P is on the ellipse. The ellipse shown here has ellipticity e 0.8.
First Definition
An ellipse is the set of points in a plane satisfying the equation
where x is the distance from the center of the ellipse of any point on the ellipse along one axis, y is the distance of the same point from the center along a perpendicular axis, and a and b are positive numbers that characterize the size and shape of the ellipse, conventionally defined so that a ≥ b. For clarity of description it is convenient to think of the x-axis as horizontal and the y-axis as vertical, though of course they can lie along any two perpendicular directions. It follows from Eq. (1) that the distance of any point on the ellipse from the center at x = 0, y = 0 satisfies
so everywhere on the ellipse
Note that where the ellipse intersects the horizontal axis we have y = 0, so x2 = a2, and therefore x = ±a; thus Eq. (1) describes an ellipse whose long diameter runs from –a to +a along the horizontal direction. Also, where the ellipse intersects the vertical axis we have x = 0, so y2 = b2, and therefore y = ±b, and Eq. (1) therefore describes an ellipse whose short diameter runs along the vertical direction, from –b to +b. (See Figure 12.) The parameter a is called the “semimajor axis” of the ellipse. It is conventional to define the eccentricity of an ellipse as
The eccentricity is in general between 0 and 1. An ellipse with e = 0 is a circle, with radius a = b. An ellipse with e = 1 is so flattened that it just consists of a segment of the horizontal axis, with y = 0.
Second Definition
Another classic definition of an ellipse is that it is the set of points in a plane for which the sum of the distances to two fixed points (the foci of the ellipse) is a constant. For the ellipse defined by Eq. (1), these two points are at x = ±ea, y = 0, where e is the eccentricity as defined in Eq. (3). The distances from these two points to a point on the ellipse, with x and y satisfying Eq. (1), are
so their sum is indeed constant:
This can be regarded as a generalization of the classic definition of a circle, as the set of points that are all the same distance from a single point.
Since there is complete symmetry between the two foci of the ellipse, the average distances and of points on the ellipse (with every line segment of a given length on the ellipse given equal weight in the average) from the two foci must be equal: = , and therefore Eq. (5) gives
This is also the average of the greatest and least distances of points on the ellipse from either focus:
Third Definition
The original definition of an ellipse by Apollonius of Perga is that it is a conic section, the intersection of a cone with a plane at a tilt to the axis of the cone. In modern terms, a cone with its axis in the vertical direction is the set of points in three dimensions satisfying the condition that the radii of the circular cross sections of the cone are proportional to distance in the vertical direction:
To Explain the World: The Discovery of Modern Science Page 31