by Colin Pask
is a manual covering the whole of mathematical astronomy as the ancients conceived it. Ptolemy assumes in the reader a knowledge of nothing beyond Euclidean geometry and an understanding of common astronomical terms; starting from first principles, he guides him through the prerequisite cosmological and mathematical apparatus to an exposition of the theory of the motion of those heavenly bodies which the ancients knew (Sun, Moon, Mercury, Venus, Mars, Jupiter, Saturn, and the fixed stars, the latter being considered to lie on a single sphere concentric with the earth) and of various phenomena associated with them, such as eclipses.1
5.1.1 Ptolemy's Starting Points
Ptolemy has a general discussion in book 1 (the Almagest is divided into thirteen books) in which he sets the scene and covers some basic assumptions. Some section headings will illustrate his thinking:
That the heavens move like a sphere.
That the earth too, taken as a whole, is sensibly spherical.
That the earth is in the middle of the heavens.
That the earth does not have any motion from place to place, either.
Ptolemy is following Aristotle and others in using a geocentric approach with the circle as the definitive geometrical element. Since the heavens must be perfect, it was long argued that motions there must also be perfect—which, to the ancient Greeks, meant uniform motion in a circle.
Thus the mathematics that Ptolemy needed is that relating to circles and spheres. In book 1, he gives the essential details, and in book 2, he tells us how to use them. The geometry of Euclid gives properties of circles, but for quantitative work, Ptolemy needs what today we call trigonometry, which allows angles and lengths to be calculated and manipulated. Ptolemy is recognized as one of the founders of trigonometry.
The basic element in Ptolemy's trigonometry is the chord of an angle, as shown in figure 5.1. If two radii in a circle form an angle α at the center, the chord of that angle is the length of line AB where A and B are the points at which those radii cut the circle. Ptolemy gives results assuming a circle of radius 60, so in terms of the modern sine function, for an angle α,
Figure 5.1. (a) The chord of angle α is given by AB, and Ptolemy assumes a circle with radius 60. (b) The quadrilateral ABCD as used in Ptolemy's theorem. Figure created by Annabelle Boag.
Other Greek mathematicians had calculated chords, but in the Almagest, we find the first truly comprehensive table giving all chords for angles ranging up to 180º and going in steps of ½º. Here are the first few entries, taken from the table in Toomer's translation:2
Arcs Chords Sixtieths
½ 0 31 25 1 2 50
1 1 2 50 1 2 50
1½ 1 34 15 1 2 50
2 2 5 40 1 2 50
2½ 2 37 4 1 2 48
The first column gives the angle in degrees, and the second column gives its chord written in terms of the base 60 system introduced by the Babylonians as we saw in chapter 2. Thus
(Now we can appreciate why Ptolemy used a circle with radius 60 in his definition of chords.)
The third column, also in base 60, tells us just how thorough and detailed Ptolemy's results are. The entries s(θ) are defined by
So s(θ) is the average increase of the chord over the half-degree interval divided by 30, and hence corresponds to the increase for one minute of arc. The values of s(θ) may be used in an interpolation calculation so that actually all angles up to 180º are effectively covered in one- minute intervals. What a brilliant achievement!
Ptolemy's definitive table would be used for centuries and by now you are probably beginning to see why he features in this book. For the interested reader, a few calculation details are given in the next section, and the books by Van Brummelen and Chabert should be consulted for further discussions.
5.1.2 Calculating Chords
I now briefly summarize how chords may be calculated according to Ptolemy. First, we note that Euclid gives results about squares, pentagons, and hexagons inscribed in circles, and using such results leads to values of the chords for angles 36º, 60º, 72º, 90º, 120º, and 180º.
Ptolemy gave his own theorem in Euclidean geometry: for the quadrilateral as shown in figure 5.1 (b), the sides and diagonals are related by
DB × AC = AB × DC + AD × BC.
Using this result with correctly chosen quadrilaterals leads to results for chords of the difference of two angles, ch(α – β), and half angles, ch(½α), that are equivalent to the well-known sine formulas:
sin(x – y) = sin(x)cos(y) – cos(x)sin(y)
2sin2(x / 2) = 1 – cos(x)
Ptolemy could then find, for example, the chord of 72 – 60 = 12º and follow that with the chords for 6º, 3º, and 1½º. Ptolemy also gives a formula for the chord of the sum of two angles. He came up with a clever scheme using inequalities to find ch(1º) (see Van Brummelen or Chabert), and then using the above results, he could construct the whole table.
5.1.3 Describing the Solar System
Ptolemy's geocentric picture of the solar system is illustrated in figure 5.2. In the Almagest, he describes how to calculate the positions of all the celestial bodies in terms of circles in various combinations. He uses the known observational data to find the model parameters, and then he produces tables that may be used to predict future astronomical observations. This is a mammoth task; I defy anyone to look through the Almagest and not be stunned by Ptolemy's achievement and the level of detail involved. (The book by Linton gives a good introduction to Ptolemy's theories. On the downside, there are claims that Ptolemy fudged some of his data—see Linton, page 70, for references to the debate on that.)
Figure 5.2. Ptolemy's geocentric picture of the solar system. Reprinted with permission, Cambridge University Press, from C. M. Linton, From Eudoxus to Einstein: A Mathematical History of Astronomy (Cambridge: Cambridge University Press, 2004).
I will give three examples of Ptolemy's astronomical calculations, beginning with the sun. Recall that the orbits must involve circles and uniform motions. Now, the sun does not move uniformly on a circle centered on the earth. To correct for that, Ptolemy displaces the circle center to a point O as shown in figure 5.3. The sun moves uniformly on that circle so the angle θ is a linear function of time. The sun's angle as viewed from the earth, θE in figure 5.2, does not vary uniformly in time, and hence the observed variable speeds of the sun in its orbit are modeled. Ptolemy must show how the “prosthaphaeresis” angle δ can be calculated so the observation angle θE = θ – δ is known. (Van Brummelen gives a detailed explanation of Ptolemy's working in chapter 2 of his book.) With the model established, Ptolemy can produce tables for the position of the sun.
Figure 5.3. Path of the sun as observed from the earth E. The path is a circle centered on O, and the sun moves uniformly around that circle. The angle θ increases linearly in time, but the angle θE observed from the earth does not. Figure created by Annabelle Boag.
My second example is Ptolemy's treatment of planetary motion. The motion of the planets (as observed from the earth) is even more complicated, and retrograde motion can occur in which the planet appears to reverse direction in its path seen against the background of the fixed stars. To handle this, Ptolemy uses the epicycle device as shown in figure 5.4.
Ptolemy stays true to the old guiding principles—uniform motion on circles—but now a more involved combination of such motions must be used. The situation for a superior planet (Mars, Jupiter, or Saturn) is shown in figure 5.4. The planet P moves uniformly around a small circle (the “epicycle”), whose center C moves in the same sense around a larger circle (the “deferent”) with center O. The center O is displaced from the earth E. There is a new point (the “equant”) Q, and it is the radial line QC that rotates uniformly, not the line OC joining the two centers. By fitting the model parameters to observed data, the properties of the planet's motion can be matched, and the motion around the epicycle accounts for the observed retrograde motion.
Figure 5.4. Orbit scheme for a superior planet (one whose orbit li
es outside Earth's). The deferent has center O, and the earth E and the equant Q are equally but oppositely displaced from O. The planet P moves uniformly around the epicycle, whose center C moves around the deferent. Figure created by Annabelle Boag.
My third example is Ptolemy's model for the motion of the moon. The motion of the moon is complicated because both the sun and the earth have a strong controlling influence over it, as Isaac Newton discovered. In fact, those complications caused Newton such trouble that allegedly he told mathematician John Machin that “my head never ached but with my studies of the moon,” and he told Halley that the theory of the moon “has broke my rest so often I will think of it no more.”3 Ptolemy's mathematical model for the motion of the moon is an epicycle model like that in figure 5.4, but now: the moon travels around the epicycle in a direction opposite to that of the center C around the deferent; the center O of the deferent itself moves around the earth on a new circle. (See figure 3.12 in Linton.) This intricate model is a tribute to his inventiveness, and fitting the various parameters would have been a most complex business. It seems clear that despite Ptolemy's first intentions, this is now well beyond the simple “uniform motion around a circle” dictate of ancient Greek philosophy, and it is remarkable that he could devise such a model to operate so well and to use in such things as eclipse calculations.
One final point: the idea of describing planetary motion in terms of a curve centered on the earth and constructed using epicycles may become clumsy, but it is mathematically sensible. An interesting commentary is given by Hanson in “The Mathematical Power of Epicycle Astronomy.”
5.1.4 An Unparalleled Achievement
As mentioned earlier, even a brief examination of the Almagest may leave the reader awestruck—and quite rightly so. It is an achievement without equal in ancient science. Of course, there were amendments and refinements made by Arab and other astronomers, but basically the Almagest was the guide to astronomy for the next 1,400 years. The first printed version appeared in Venice in 1515 CE. The most useful tables in the Almagest were edited and published separately in the Handy Tables. The Handy Tables were translated into many languages and had a larger circulation than the Almagest itself; in historical terms “their longevity, wide distribution and influence among astronomers worldwide mean that Ptolemy's Handy Tables can justifiably claim to be the first mass-produced mathematical table.”4
Surely no one could doubt that the list of candidates for the title of the great calculations must include calculation 13, Ptolemy's Almagest.
5.2 THE GREAT STEPS
The theoretical astronomy discussed above was based on two key ideas: the earth is the center of the solar system (geocentric model); and the mathematics to be used must involve uniform motion in circular paths. It is clear that Ptolemy was struggling to satisfy those requirements, and ingenious devices were devised to build them into his mathematical models. Astronomy could make giant steps forward only when those key ideas were challenged and overthrown.
5.2.1 The Sun Becomes Supreme
There were ancient Greeks, like Aristarchus, and later mathematicians, for example Nicole Oresme (1320–1382), who suggested that the sun, rather than the earth, should be at the center of the solar system and that the earth is in motion. However, it is in some ways easier to match observations made on Earth using a mathematical model centered on the earth, and we have seen that Ptolemy did just that with considerable success. But there are problems with a geocentric model, and after many centuries it was becoming evident that a more accurate description of the solar system was needed. The great shift to a heliocentric viewpoint was made by the Polish astronomer Nicholas Copernicus (1473–1543) in his On the Revolutions of the Heavenly Spheres, the published form of which he saw on his deathbed. The Earth now rotated and moved on an orbit like the other planets.
A model with both the earth and the planets orbiting the sun gives a simple explanation of a planet's retrograde motion (see figure 5.1 in Linton). It also explains why we see phases for Venus (famously observed by Galileo) as shown in figure 5.5. The hope is that the mathematical details of the orbits will also be simpler so that more accurate tables may be produced. But Copernicus still held to the ancient Greek mandate that motion in the heavens must be perfect and involve only uniform motion in circles. The calculations of orbits by Copernicus still used devices like epicycles following the same pattern as Ptolemy. Linton suggests that “Copernicus might well be described as the last of the ancients, a spiritual companion of Aristarchus, Hipparchus and Ptolemy.”5
Figure 5.5. Why we see phases for Venus. From Johannes Kepler, Epitome Astronomiae Copernicanae (1621).
Copernicus knew that he lacked direct proof that the earth was indeed in motion, and he was aware that without it, he would encounter the hostility of the Catholic Church to any suggestion that the earth should lose its preeminent role as the center of the universe. The Lutheran theologian Andreas Osiander took care of the printing of Copernicus's book, and without the author's knowledge, he inserted these highly significant words into a preface:
For these hypotheses need not be true or even probable. On the contrary, if they provide a calculus consistent with observations, that alone is enough…. For this art, it is quite clear, is completely and absolutely ignorant of the causes of the apparent nonuniform motions. And if any causes are devised by the imagination, as indeed very many are, they are not put forward to convince anyone that they are true, but merely to provide a reliable basis for computation.6
Copernicus had updated Ptolemy's approach by changing from an Earth- to a sun-centered solar system, and Osiander is asking readers to forgive him that step because he is only trying to construct the best mathematical model. (Of course, we do not know Copernicus's reaction to Osiander's intrusion since he was dying when the book came into his hands.)
5.2.2 Enter the Game Changer: Johannes Kepler
Occasionally in history there is someone who changes the direction of science is an important and profound way. Such a man was Johannes Kepler (1571–1630). He was born in Germany and educated at Tübingen, where his university studies covered astronomy, mathematics, astrology, and theology. His intention was to be a clergyman, but through direction and opportunity, he took on a career as mathematician and astronomer. He remained deeply religious and something of a mystic. Kepler traveled widely in Europe and spent periods in Graz and Prague. His eventful and colorful life (at one time he had to defend his mother against charges of witchcraft) is described in the Dictionary of Scientific Biography article by Gingerich, who gives a comprehensive bibliography covering everything about Kepler and his work. The proceedings of a conference marking the 400-year anniversary of Kepler's work were edited by Arthur and Peter Beer to create an encyclopedic, thousand-page reference and additional resource for the Kepler addict.
Kepler was an ardent Copernican, but his work was driven by a new approach: he wished to go beyond the mathematical models of the astronomer to appreciate the underlying mechanisms sought by the physicist. In 1596, he published Mysterium Cosmographicum with his intentions revealed in the introduction:
And there were three things above all for which I sought the causes why it was this way and not another—the number, the dimensions, and the motions of the orbs.7
In summary, Kepler still used mathematical models, but now the search for underlying physical causes was to be used to explain and motivate those models.
In Kepler's time there were the five planets known in ancient times plus the new Copernican planet Earth. The natural questions for Kepler were: Why just six planets? And why are they spaced out as we observe them? These are the mysteries he tackled in Mysterium Cosmographicum. In a demonstration of his brilliance and inventiveness, Kepler noted that Euclid concluded the Elements by showing that there can be only five regular polyhedrons—or perfect or Platonic solids, as they are sometimes called. (The faces of a regular polyhedron are all identical and must be regular polygons. See figure 5.6.) Furthermore
, Kepler found that those five solids could be nested inside one another with a sphere drawn at each interface, see figure 5.6. He then suggested that the planets are arranged on these spheres and calculating the resulting spacing from the geometrical arrangement, he found quite reasonable agreement (within 5 percent) with the accepted planetary distances.
Figure 5.6. The five perfect polyhedrons, and Kepler's system of nested polyhedrons and accompanying spheres. From Johannes Kepler, Mysterium Cosmographicum (1596).
Kepler's result is stunning; a mathematical classification result in geometry is used to explain why the number of planets is limited and why they are spaced out as in the solar system. Of course, the discovery of more planets spoils Kepler's scheme, but it remains as an impressive and ingenious calculation.
5.2.3 Revolutionary Advances
The Danish astronomer Tycho Brahe (1546–1601) recorded an enormous collection of data and raised the accuracy of observational astronomy to a new level. Kepler met Brahe in 1600 and took over as imperial mathematician when Brahe died. Kepler was charged with accounting for the orbit of Mars and the production of new astronomical tables, the Rudolphine Tables, named for the Emperor Rudolph. So began the battle with Mars.
Kepler's work using the data on Mars is one of the epic calculations involving great dedication, supreme technical abilities, and a new and exemplary respect for scientific principles. He began with the usual Copernican orbits with a range of variations as he tried to fit Brahe's superb data. For each new model, Kepler had to use data to fit the model parameters, and then he tested it against other data. The thoroughness of his work and the level of accuracy he demanded are remarkable. One point must be emphasized: Kepler refused to be satisfied with models resulting in small errors that others at that time would readily ignore. Also, unlike most scientists (particularly present-day ones), Kepler documented his failings and tortuous path, although few readers would ever wish to follow it all in detail; in fact, at one point he wrote: “if you are wearied by this tedious procedure take pity on me who carried out at least seventy trials.”8 The story can be read in the references to Gingerich (his Physics Today article is beautifully written and illustrated), and also in Linton, Thurston, and in Koyré's major study. It is hard for us today to imagine how one man could produce the enormous number of detailed calculations like those taken from one of Kepler's notebooks for figure 5.7.