Ptolemy’s Geography and Astronomy - Greek Mathematics From 500 BCE to 500 CE - A brief course - The history of mathematics

The history of mathematics: A brief course (2013)

Part III. Greek Mathematics From 500 BCE to 500 CE

Chapter 17. Ptolemy's Geography and Astronomy

The path away from the metric-free geometry of Euclid, Archimedes, and Apollonius was opened by Heron, Ptolemy, and other geometers who lived during the early centuries of the Roman Empire. Ptolemy's Almagest is an elegant arithmetization of some basic Euclidean geometry applied to astronomy. As in the work of Heron, proportions provided the key to arithmetizing triangles and circles in a way that made a computational geometric model of the motions of the heavenly bodies feasible. In the Almagest, computations using a table of chords are combined with rigorous geometric demonstration of the relations involved. Ptolemy (ca. 85–ca. 165), whose very name shows his Alexandrian origins even though he lived in Rome, shows that he is well acquainted with the geometry and astronomy of his day. But he studied the earth as well as the sky, and his contribution to geography is also a large one, and also geometric, although less computational than his astronomy. We shall begin with the geography, for which an annotated translation exists (Berggren and Jones, 2000).

17.1 Geography

Ptolemy was one of the first scholars to look at the problem of representing large portions of the earth's surface on a flat map. His data, understandably very inaccurate from the modern point of view, came from his predecessors, including the astronomers Eratosthenes (276–194) and Hipparchus (190–120) and the geographers Strabo (ca. 64 BCE –24 CE) and Marinus of Tyre (70–130), whom he followed in using the now-familiar lines of latitude and longitude. These lines have the advantage of being perpendicular to one another, but the disadvantage that the parallels of latitude are of different sizes. Hence a degree of longitude stands for different east–west distances at different latitudes.

As an empirical science, geography depends intimately on these two coordinates, whose empirical foundations are very different. Latitude is relatively easy to determine. If you look at any known star when it is directly above your local north–south line, you can measure its altitude above the southernmost point on the horizon. Then if you add 90img and subtract the known declination of that star on the celestial sphere, the difference will be your geographical latitude.1 Longitude is much more difficult to determine, since the same stars will pass overhead at the same local time on a given day at all points having the same latitude. To determine longitude, it is necessary to single out one meridian, called the prime meridian, for use as the origin. By universal convention, that meridian is the one through Greenwich, England. You can work out your longitude east or west of Greenwich provided you know what time it is in Greenwich at any moment of your choosing. The difference between your local solar time and the local solar time at Greenwich is your longitude. (Each hour of time difference represents 15 degrees of longitude.) In these days of global positioning systems, terrestrial longitudes are known precisely, and there is no reason to doubt the data. But in the days before the instant communication of radio, and in the absence of a clock that would keep perfect time while being transported over a considerable portion of the earth's surface, longitude was very difficult to determine. What was needed was an event that could be observed from all over the earth simultaneously, to be followed by a comparison of the local times at which they occur. One such event occurred at the time of the Battle of Arbela, which was mentioned in Chapter 11. According to Pliny the Elder (23–79) in his Natural History, Book 2, Chapter 72)2:

We are told that at the time of the famous victory of Alexander the Great, at Arbela, the moon was eclipsed at the second hour of the night, while, in Sicily, the moon was rising at the same hour.

This would mean that Sicily is about 30img west of Arbela, and in fact Sicily straddles the meridian at 14img E while Arbela is at 44img E.

Such events are rare, and comparisons of them are hard to coordinate. The search continued for a large celestial clock. The phases of the moon would seem to be an obvious one, which everyone on the same side of the earth can observe at once. However, they change too slowly to allow precise measurement. When he discovered four moons of Jupiter, Galileo realized that their configuration, which changed fairly rapidly, could be used as the clock he wanted. Once again, however, measurements could not be made with sufficient precision to be of any use. Not until a durable and accurate spring-wound clock was created could this problem be effectively solved. Ptolemy was forced to rely on travel times over east–west routes to determine relative longitudes.

Ptolemy assigned latitudes to the inhabited spots that he knew about by computing the length of daylight on the longest day of the year. This computational procedure is described in Book 2, Chapter 6 of the Almagest, where Ptolemy describes the latitudes at which the longest day lasts img hours, img hours, and so on up to 18 hours, then at half-hour intervals up to 20 hours, and finally at 1-hour intervals up to 24. Although he knew theoretically what the Arctic Circle is, he didn't know of anyone living north of it, and took the northernmost location on the maps in his Geography to be Thoúlimg, described by the historian Polybius around 150 BCE as an island six days sail north of Britain that had been discovered by the merchant–explorer Pytheas (380–310) of Masillia (Marseille) some two centuries earlier.3 It has been suggested that Thoúlimg is the Shetland Islands (part of Scotland since 1471), located between 60img and 61img north; that is just a few degrees south of the Arctic Circle, which is at 66img 30'. It is also sometimes said to be Iceland, which is on the Arctic Circle, but west of Britain as well as north. Whatever it was, Ptolemy assigned it a latitude of 63img, although he said in the Almagest that some “Scythians” (Scandinavians and Slavs) lived still farther north at img. Ptolemy did know of people living south of the equator and took account of places as far south as Agisymba (Ethiopia) and the promontory of Prasum (perhaps Cabo Delgado in Mozambique, which is 14img south). Ptolemy placed it 12img 30' south of the Equator. The extreme southern limit of his map was the circle 16img 25' south of the equator, which he called “anti-Meróimg,” since Meróimg (a city on the Nile River in southern Egypt) was 16img 25' north.

Since he knew only the geography of what is now Europe, Africa, and Asia, he did not need 360img of longitude. He took his westernmost point to be the Blessed Islands (possibly the Canary Islands, at 17img west). That was his prime meridian, and he measured longitude out to 180img eastward from there, to the Simgres,4 the Chinese (Sínai), and “Kattígara.” According to Dilke (1985, p. 81), “Kattígara” may refer to Hanoi. Actually, the east–west span from the Canary Islands to Shanghai (about 123img east) is only 140img of longitude. Ptolemy's inaccuracy is due partly to unreliable reports of distances over trade routes and partly to his decision to accept 500 stades, about 92.5 km—a stade is generally taken to have been 185 m—as the length of a degree of latitude. The true distance is about 600 stades, or 111 km.5 We are not concerned with the units in Ptolemy's geography, however, only with its mathematical aspects.

The problem Ptolemy faced was to draw a flat map of the earth's surface spanning 180img of longitude and about 80img degrees of latitude, from 16img 25' south to 63img north. Ptolemy described three methods of doing this, the first of which we shall now discuss. The latitude and longitude coordinates of the inhabited world (oikuménimg) known to Ptolemy represent a rectangle whose width is img of its length. Ptolemy did not represent parallels of latitude as straight lines; he drew them as arcs of concentric circles while keeping the meridians of longitude as straight lines emanating from the common center, representing the North Pole. Thus, he mapped this portion of the earth into the portion of a sector of a disk bounded by two radii and the arcs they cut off on two circles concentric with the disk. As shown in Fig. 17.1, the first problem was to decide which radii and which circles are to form these boundaries. Ptolemy recognized that it would be impossible in such a map to place all the parallels of latitude at the correct distances from one another and still get their lengths in proportion. He decided to keep his northernmost parallel, through Thoúlimg, in proportion to the parallel through the equator. That meant these two arcs should have a ratio of about 9:20—more precisely, cos (63img) in our terms, which is 0.45399. Since there would be 63 equal divisions between that parallel and the equator, he needed the upper radius x to satisfy x : (x + 63) : : 9 : 20. Solving this proportion is not hard, and one finds that x = 52, to the nearest integer. The next task was to decide on the angular opening. For this principle he decided, like Marinus of Tyre, to get the parallel of latitude through Rhodes in the correct proportion. Since Rhodes is at 36img latitude, the length of half of the parallel of latitude through it amounts to about img of the 180imgarc of a great circle, which is about 145img. Since the radius of Rhodes must be 79 (27 great-circle degrees more than the radius of Thoúlimg), he needed the opening angle of the sectors θ to satisfy θ : 180img : : 146 : 79π, so that θ ≈ 106img. After that, he inserted meridians of longitude every one-third of an hour of longitude (5img) fanning out from the North Pole to the Equator.

Continuing to draw the parallels of latitude in the same way for points south of the Equator would lead to serious distortion, since the circles in the sector continue to increase as the distance south of the north pole increases, while the actual parallels on the earth begin to decrease at that point. The simplest solution to that problem was to let the southernmost parallel at 16img 25' south have its actual length, then join the meridians through that parallel by straight lines to the points where they intersect the equator. Once that decision was made, he was ready to draw the map on a rectangular sheet of paper. He gave instructions for how to do that: Begin with a rectangle that is approximately twice as long as it is wide, draw the perpendicular bisector of the horizontal (long) sides, and extend it above the upper edge so that the portion above that edge and the whole bisector are in the ratio 34img : 131img, 25'. In that way, the 106img arc through Thoúlimg will begin and end just slightly above the upper edge of the rectangle, while the lowest point of the map will be at the foot of the bisector, being about 80 units below the lowest point on the parallel of Thoúlimg, as indicated by the dashed line in Fig. 17.1.

Although at first sight, this way of mapping seems to resemble a conical projection, it is not that, since it preserves north–south distances. It does a tolerably good job of mapping the parts of the world for which Ptolemy had reliable data.

Figure 17.1 Ptolemy's first method of mapping.

img

17.2 Astronomy

Ptolemy's astronomy, for which there are good sources in English (Toomer 1984a,b, Jones 1990) was much more geometrical than his geography. It established metrical relations among chords and arcs of circles by means of which it was possible to give latitude and longitude coordinates (what are now called declination and right ascension, respectively) for all the stars and planets (including the sun and moon among the latter). These coordinates were imposed on what is called the celestial sphere, which is a representation of all the stars as if they were stuck on a sphere whose center was at the center of the earth. It is shown in Fig. 17.2. The circle labeled ecliptic in that figure is the path that the sun follows as it moves through the fixed stars, making one circuit per year. It crosses the celestial equator moving from south to north on March 20 (rarely, on March 21). That intersection of the ecliptic and equator is called the vernal equinox. It provides a natural prime meridian on the celestial sphere, from which right ascension can be measured. (There is no such natural prime meridian on the earth, and the one actually used—through Greenwich—is a human convention.) Since it was impossible in Ptolemy's day to tell the distances to the fixed stars, what we call the radial coordinate in three-dimensional spherical coordinates was irrelevant. Using this celestial sphere, with the vernal equinox serving as origin, one could assign permanent locations to all the fixed stars, leaving only seven celestial bodies known to Ptolemy (sun, moon, Mercury, Venus, Mars, Jupiter, and Saturn) as “wandering” stars whose coordinates were constantly changing among the fixed stars. The problem for geocentric astronomy was to express that wandering as a combination of simple circular motions.

Figure 17.2 The celestial sphere.

img

As with his geography, Ptolemy benefited from data assembled over a long period of time, namely observations of the positions of various planets, times of eclipses, and other celestial phenomena, made at various places in the Mediterranean world, including Mesopotamia, over the preceding 800 years. To fit all these data to observation, he used a system known as epicycles and/or eccentrics. An epicycle is a uniform motion about the center of a circle that is itself moving uniformly around a second circle, called the deferent. An eccentric is a uniform motion in a circle, but observed from a point not at the center of the circle. Either of these devices can be used to account for the observed variable speeds of a heavenly body around the celestial sphere.

17.2.1 Epicycles and Eccentrics

The fact that a uniform motion along a circle viewed from an eccentric point is exactly the same as a uniform motion along an epicycle combined with a uniform motion of the epicycle can be seen in Fig. 17.3, in which the center of a small circle (the epicycle) moves along the larger circle (the deferent) in such a way that the angle through which a point on the epicycle has rotated clockwise relative to the line joining the center of the epicycle to the center of the deferent equals the angle through which the center of the epicycle has moved counterclockwise along the deferent, measured from a fixed diameter of the deferent. Because OAB' C' is a parallelogram, the angle B' AB that an observer at A measures between the original line AC and the line AB' to the center of the epicycle is exactly the same as the angle that an observer at the center O of the deferent measures between the original line AC and the line to the point C' on the epicycle. Ptolemy demonstrated this equivalence in Section 3 of Book 3. He also considered the possibility that the sun could rotate on its epicycle in the same direction that the epicycle moves around the earth, pointing out that in that case, the most rapid motion would be at apogee (when the sun is farthest from the earth) and the slowest at perigee (when the sun is closest to the earth). In fact, the most rapid motion is at perigee (or perihelion, as we call it, using the heliocentric system). That occurs within a day or two of January 3 each year.6

Figure 17.3 Equivalence of epicycles and eccentrics.

img

With either model, eccentric or epicyclic with the same angular velocity, no retrograde motion will be observed. The advantage of the epicycle is that retrograde motion can be accounted for by taking the angular velocity on the epicycle greater than that of the epicycle on the deferent. If, for example, the velocity of the planet on the epicycle is twice that of the epicycle on the deferent, retrograde motion will be observed if the radius of the epicycle is more than half the radius of the deferent.

17.2.2 The Motion of the Sun

Since we do not have space to discuss the complexities of the Almagest, we shall confine our discussion to a brief sketch of the sun's motion. In this case, the epicycle in Fig. 17.3 can be thought of as showing the approximate positions C and C' of the sun starting in early July (at C) and then at a somewhat later time, around mid-September (at C'). In the eccentric hypothesis, the sun is at B and B' respectively at those times and moving at a uniform rate along the deferent, but the observer (on the earth) is located at A. Whether we imagine an observer at O (the center of the deferent) observing the point C on the epicycle, or an observer at A (displaced from the center of the deferent) observing the point B on the deferent, makes no difference, since both will observe exactly the same amount of rotation at any given time, namely the angle C' OC or the angle B' AB.7

To summarize: The mean position of the sun (B) moves at a constant rate around the deferent, and the deviation from the mean is accounted for either by saying that the earth isn't at the center of the deferent, or that the sun is revolving around its mean position on the epicycle, again at a uniform rate. Either assumption allows the actual motion to be uniform while its appearance to terrestrial observers is not uniform.

The single-epicycle, or eccentric, model is well suited for a comparatively simple motion such as that of the sun. The path of the sun among the stars is the ecliptic, which for our purposes is regarded as a fixed circle. Its motionalong this path, however, is not at a uniform angular rate. It moves most slowly when passing through the constellation Gemini, which astronomers and astrologers refer to as the House of Cancer. (When the houses of the Zodiac were originally established, the House and the constellation of the same name coincided. Because of precession of the equinoxes, they are now about one month out of phase.) The summer solstice in Hellenistic times was in the constellation Cancer, so that the slowest motion of the sun occurred before the solstice, in late May. Nowadays the apogee (aphelion in the heliocentric system) is reached in early July, shortly after the summer solstice (the northernmost point on the ecliptic). Hipparchus placed the apogee about 24img before the summer solstice. Using this information and the fact that (in his day), spring was img days long while summer was img days long, Ptolemy managed to fit the sun's motion by using an epicycle and deferent whose radii were in the ratio of 1 : 24. This ratio, briefly improved upon by Copernicus before Kepler banished circles altogether in favor of ellipses, gave good agreement with observation. Using that scheme and fitting the data to the appropriate dates in 2010, one can obtain the following table of right ascensions of the sun at 30-day intervals throughout the year. The third column of the table gives the values computed by modern astronomy, and the last column shows the amount by which “Ptolemy's” (actually, our) predicted values fall short of the modern values, which is always less than 4' of arc, or one degree.8

img

17.3 The Almagest

There is insufficient space here to describe Ptolemy's entire treatise, and in any case our primary concern is with its mathematical innovations. To make geometric astronomy work, Ptolemy developed a subject that resembles what we now call spherical trigonometry, extending earlier work by Hellenistic mathematicians such as Menelaus of Alexandria.

17.3.1 Trigonometry

The word trigonometry means triangle measurement, but angles are generally measured in terms of the amount of rotation they represent, that is, in terms of the ratio of the length of the arc they subtend to the circumference of the circle containing the arc. That is the context in which Ptolemy developed the subject. It is essentially the study of the quantitative relations between chords and arcs in a circle.

In a system that is still basically the standard one, Ptolemy divides the circumference into 360 equal parts, and measures angles in terms of those parts, that is, in degrees (sometimes half-degrees). The basic problem of trigonometry, from this point of view, is to determine the length of the chord subtended by a given arc and vice versa. To this end, following the Babylonian sexagesimal system, Ptolemy uses img of the radius of the circle as the unit of length for chords in a given circle. The effect of this technique is that when two circles intersect, their common chord must be expressed in two different ways, in terms of the two radii. This procedure leads to constant scaling of lengths, and is apt to provoke an impatient reaction from the modern reader. Cumbersome though it was, however, it worked and enabled Ptolemy to give an accurate quantitative description of celestial motions.

17.3.2 Ptolemy's Table of Chords

The computation of the table of chords used by Ptolemy is an interesting exercise in numerical methods. The natural approach would be to start with a central angle whose chord is known (say, 60img, for which the chord equals the radius of the circle), then use half-angle formulas to compute the chord of 30img, 15img, 7img 30', and so on, until the desired tabular difference is achieved, after which one would build up the table in these intervals using the addition formulas for the trigonometric functions.9 Ptolemy's approach is like this, but he does the computations very elegantly, using what is now called Ptolemy's theorem: In a quadrilateral inscribed in a circle, the rectangle on the diagonals equals the sum of the rectangles on the two pairs of opposite sides. To prove this theorem, draw a line BE from the vertex B to the diagonal AC such that ∠ABE = ∠ DBC, as in Fig. 17.4. Hence ∠EBC = ∠ ABD. Therefore, since angles BAC and BDC are both inscribed in the same arc, triangles ABE and DBC are similar. For the same reason, triangles EBC and ABD are similar. It follows that AB · CD + BC · AD = AE · BD + EC · BD = AC · BD.

Figure 17.4 Ptolemy's theorem.

img

Ptolemy's theorem makes it possible to express the chord on the difference of two arcs in terms of the chords on the individual arcs. Given three points on a circle, say A, B, and C, take point D diametrically opposite one of the points, say A (see Fig. 17.5). If the chords AC and AB are given, draw the diameter AD and the chords BC, DB, and CD. The chord AD is known, being the diameter of the circle (hence equal to 120 of Ptolemy's units). Then DB and DC can be computed using the Pythagorean theorem from the diameter and the given chords, since an angle inscribed in a semicircle is a right angle. Hence in the inscribed quadrilateral ABCD both diagonals and all sides except BCare known, and so BC can be computed.

Figure 17.5 The chord of 12img.

img

Ptolemy used this theorem to construct a table of chords of angles at half-degree intervals. He began with a regular decagon inscribed in a circle. The central angles subtended by the sides of this decagon are each equal to 36img. Because of the compass-and-straightedge construction of this figure, its side can be expressed as img, where r is the radius, which is 60 standard units according to Ptolemy. Instead of repeatedly bisecting this angle, however, Ptolemy adopted an indirect strategy to find the chord of a smaller angle without having to extract so many square roots. He used the fact that the side of the regular pentagon inscribed in a circle (the chord of 72img) is known from Euclid's Elements, Book 13, Proposition 10 to be the hypotenuse of the right triangle whose legs are the radius of the circle and the side of the inscribed regular decagon. Thus this chord is

equation

which, given that r = 60, is 70.5342302751. . .. In Ptolemy's sexagesimal notation, this number is given to the nearest second as 70 ; 32, 3. Since the chord of 60img is obviously r, which is 60 standard units, one can then use Ptolemy's theorem to compute the chord of 72img − 60img = 12img. In order to apply this theorem, we first need to get the chords on the arcs of 120img and 108img supplementary to these two angles, as shown in Fig. 17.5. By the Pythagorean theorem, we get

equation

Thus, the chord of 12img is

equation

and given that r = 60, this becomes approximately 12.5434155922. . . . Ptolemy gave it as 12 ; 32, 36.

We have given this computation in language that is more symbolic than Ptolemy's. He always wrote 60 where we have written r, and he had no symbol for the square root. He would first write down the square whose root is to be taken, as a number rather than an expression, and then write down the square root, again as a number.

Ptolemy then showed how to compute the chord of half an angle if the chord of the angle is known. In this way he was able to compute successively the chords of 6img, then 3img, then 1img 30', and finally 0img 45'. He found that, to three sexagesimal places, the chord of 1img 30' is 1 ; 34, 25 and the chord of 0img 45' is 0 ; 47, 8. The ingenious idea of starting from a 72img angle, rather than the more natural 60img angle, allowed Ptolemy to reach angles less than 1img while minimizing the roundoff error caused by approximating square roots.

Ptolemy's construction of his table misses the important angle of 1img. This gap is not accidental. All the angles whose chords can be found by his strategy can be constructed with compass and straightedge, but a 1img angle is not constructible with these instruments alone. To estimate the chord of 1img, Ptolemy combined the two chords on each side of 1img, namely 1img 30' and 0img 45' with a useful approximation theorem: The ratio of the larger of two chords to the smaller is less than the ratio of the arcs they subtend. We have already encountered a theorem similar to this, but not quite identical, in connection with the work of Zenodorus on the isoperimetric problem. (Compare Fig. 16.1 of Chapter 16 with Fig. 17.6 of the present chapter.)

Figure 17.6 A fundamental inequality from Ptolemy's Almagest.

img

In other words, the ratio of a larger chord to its arc is less than the ratio of a smaller chord to its arc. In still other words, the ratio of chord to arc decreases as the arc increases. In our own language, using radian measure for angles, the chord of an arc of length is 2r sin (θ/2). This theorem says that the ratio img decreases as ϕ increases. Because of this proposition, the chord of 1img is smaller than four-thirds of the chord of 0img 45', and larger than two-thirds of the chord of 1img 30'. If we treat Ptolemy's two approximations as exact, we find that the chord of 1img is less than 1 ; 2, 50, 40 and larger than 1 ; 2, 50. Ptolemy truncated the first of these to 1 ; 2, 50. He then wrote (what is logically absurd) that “the chord of 1img has been shown to be both greater and less than the same amount.” But we know what he means.

Thus Ptolemy had established that the chord of 1img is approximately 1 ; 2, 50 units when the radius is 60 units. Then, using his half-angle formula, he found the chord of 0img 30' to be 0 ; 31, 25, after which he was able to construct the table of chords for angles at half-degree intervals.

The table of chords makes it possible to solve right triangles, in particular, to find the angles in such a triangle when given the ratio of two of its sides. In astronomy, however, one is always using angular coordinates on a sphere, since both the sides and angles of a spherical triangle are given as angles. It would be clumsy always to have to introduce plane triangles in order to find the parts of spherical triangles, and so Ptolemy included certain relations among the parts of spherical triangles as lemmas. These are not the laws of cosines and sines now used in spherical trigonometry, but rather two theorems that had been published half a century earlier in a work called Sphaerike by Menelaus of Alexandria. With these relations it is possible to solve such problems as finding which portion of the ecliptic rises simultaneously with a given portion of the celestial equator.

With this mathematical equipment and a wealth of observational data, Ptolemy was able to apply the theoretical methods invented by earlier astronomers. The 12 books of the Almagest became the standard astronomical treatise in the Middle East and Europe until the sixteenth century. The details are too complicated to summarize, and we shall have to leave the reader with just the sample given above for the motion of the sun over a single year.

Problems and Questions

Mathematical Problems

17.1. Use Fig. 17.6 to show that the ratio of a larger chord to a smaller is less than the ratio of the arcs they subtend, that is, show that BΓ : AB is less than img, where ΔZ is the perpendicular bisector of AΓ. (Hint: BΔ bisects angle ABΓ.) Carry out the analysis carefully and get accurate upper and lower bounds for the chord of 1img. Convert this result to decimal notation, and compare with the actual chord of 1img which you can find from a calculator. (It is img.)

17.2. In modern language, the chord of an arc α can be expressed as img, where d is the diameter of the circle. Referring to Fig. 17.7, show that Ptolemy's theorem is logically equivalent to the following relation, for any three arcs α, β, and γ of total length less than the full circumference.

equation

Figure 17.7 Ptolemy's theorem.

img

17.3. Compute the chord of 6img in two different ways: (1) by expressing 6img as the difference of a 36img arc and a 30img arc, whose chords are known to be img and img; (2) by expressing it as the chord of the difference of 12img and 6img.

Historical Questions

17.4. How did the ancients determine geographical latitude?

17.5. Why was geographic longitude more difficult to determine than latitude?

17.6. Out of what mathematical and observational data did Ptolemy construct his astronomical treatise?

Questions for Reflection

17.7. Does the success of Ptolemy's Almagest vindicate Plato's conviction that the key to understanding the material world was to connect it with an ideal world of abstractions (ideas or forms) that are perceived with the mind rather than the senses?

17.8. In Ptolemy's system, the occasional retrograde motion of, say Mars, is explained by its motion on the epicycles, whose deferents are at or near the center of the earth. Now, retrograde motion of the outer planets—westward rather than eastward on (or near) the ecliptic—is observed in the time interval from just before to just after a planet is in opposition to the sun, that is, 180img opposite the sun on the celestial sphere. The fitting of epicycles for any planet must therefore take account of the position of the sun, which itself never undergoes retrograde motion. Surely, one would think, these considerations suggest that the planets are more closely tied to the sun than to the earth, and heliocentric astronomy would be much simpler than geocentric. And in fact, Ptolemy considered this hypothesis, which had been proposed by the astronomer Aristarchus of Samos (ca. 310–ca. 230), but he rejected it on physical grounds. Why did it take another 1500 years for this hypothesis to be revived and become the cornerstone of modern astronomy?

17.9. Ptolemy used a wheel submerged in water up to its axle in order to determine the refraction of light in passing from air into. Based on his observations he gave the following table of the angles of refraction for angles of incidence of 10img, 20img,. . ., 80img. (These are the angles the ray makes with the vertical as it enters the water. The angles of refraction are the angles the ray makes with the vertical after entering the water.) The third column gives the values computed from Snell's Law. Since the ratio of the velocities is about 4 : 3 for light in air and light in water, Snell's law says that sin (ϕ) = 3 sin (θ)/4, where is θ is the angle of incidence and ϕ is the angle of refraction.

equation

Since physical scientists, and Ptolemy in particular, are known to have manipulated their data to fit a theory, does this table indicate any such manipulation?

Notes

1. For the definitions of declination and local altitude, see Section 17.2 below.

2. Arbela is modern Erbil, Iraq, but the battle took place some 80 km distant from it. At that battle in 331 BCE, Alexander defeated the Persian King Darius and effectively put an end to the Persian Empire.

3. The Latin idiom ultima Thule means roughly the last extremity.

4. The Simgres were a Hindu people known to the Greeks from the silk trade.

5. One measurement of the length of a degree available to Ptolemy was that of Eratosthenes, who used shadow lengths at the summer solstice in Syene and Alexandria, Egypt to conclude (correctly) that Alexandria is about 7imgnorth of Syene. (It is actually about 3img west as well.) Eratosthenes gave the distance between the two cities as 5000 stades. The actual distance is about 730 km. These figures are inconsistent with the length of a stade given above and the number of stades in a degree, but Ptolemy had other sources to reconcile with Eratosthenes.

6. Perihelion for the center of gravity of the earth–moon system occurs on January 3; but because the two bodies rotate about a common center of gravity, that is not necessarily the day when the center of the earth is at perihelion.

7. It is impossible to observe the distances to the stars with the unaided eye, so that the only things we can actually measure are the angles between our lines of sight to two different stars. Thus, the distances (radii of the epicycles) can be anything they have to be to account for the angles we actually observe.

8. A minute in this context is a minute of time—one-sixtieth of an hour, and an hour is 15 degrees.

9. In fact the algorithm by which hand calculators evaluate the trigonometric functions works roughly along these lines. Certain values are hard-wired into the calculator and others are computed by application of the addition formulas.