But I admired even more what Eames achieved on the flight back home. He began thinking very hard about how the Copernican Sun-centered model for planetary motions differed from the traditional Ptolemaic Earth-centered scheme. Each arrangement required two circles to explain the motion of Mars as seen from the Earth. In the Copernican model the Earth's orbit and Mars's orbit each circled the Sun. But in the geocentric setup the Earth was at rest in the center, with a large deferent circle (i.e., the carrying circle) going around it. The second circle (called the epicycle) contained the planet Mars and rode on the deferent. Charles knew that each arrangement had to give the same results because each system had to represent the same observations, and he wanted to make a dynamic model to show it. By the time he flew into Los Angeles, he had the device all sketched out, complete with the behind-the-scenes linkages (accomplished with bicycle chains). The front face of his model looked like this, with the straight rods representing the observational line of sight from the Earth to Mars in each arrangement:
Ptolemy's Geocentric System
Copernicus' Heliocentric System
When he got to Venice, he turned the plans over to his shop technician, and a few days later he had a working model. In the exhibition, which opened in December of 1972, the Eames machine ran continuously without default for something like six months (plate la). As the circles turned, the rods, representing the observed line of sight to Mars, always remained parallel. Each time Mars came on the inner side of the epicycle, the combined counterclockwise motions of the deferent and epicycle caused the geocentric rod to briefly swing clockwise, the so-called retrograde motion. Whenever that happened, in the heliocentric model the faster-moving Earth was always nearest Mars and bypassing it, so the heliocentric rod remained in perfect tandem with the geocentric rod. It was a brilliant demonstration of the equivalence of the two systems, and what worked for Mars would work for each of the other planets. Over the years several of my students managed to simulate Eames' model on their computer screens, but no one was ever able to figure out how to do it with bicycle chains.
* The Frauenburg cathedral library must have owned a copy of Copernicus' De revolutionibus, but no such copy has been located.
* The Latin spellings of their names are Nicolaus and Andreas. I generally refer to the astronomer as Nicolaus except when discussing his adolescent and student days.
† Scholars speculate that frugal Copernicus took his degree at Ferrara, where he had not actually studied, because he knew no one there and hence was spared the expenses of a lavish graduation party.
* Some years after the Copernican Quinquecentennial I got an earnest phone call from a Chicago book dealer asking for an evaluation of the deluxe facsimile, which was by then out of prinr and worth some hundreds of dollars. It didn't take long for me to guess what had happened. A well-meaning but relatively unsophisticated Polish-American citizen of Chicago was presenting a facsimile to the Adler Planetarium, and he wanted to take a tax deduction. The only problem was that he thought he was giving the planetarium Copernicus' original manuscript.
* The University of Louisville generously lent its Narratio prima. The other four American copies of rhis edition were at Harvard, Yale, the Burndy Library (now the Smithsonian's Dibner Library), and in the private collection of Robert Honeyman in California. Since that time the Honeyman copy has been auctioned and has become the only located copy in Italy. Meanwhile rwo previously unrecorded copies have come to America, one to a private collection and one to the Linda Hall Library in Kansas City.
Chapter 4
THE LENTEN PRETZEL
AND THE EPICYCLES MYTH
QUIETLY SLEEPING on the title page of Derevolutionibus is a Greek epigram reading Ageometretos medeis eisito (Let no one untutored in geometry enter here), by legend the motto on the gate of Plato's academy in ancient Athens. Any potential buyer who could decipher the Greek would most likely have been able to handle the geometry. Nevertheless, I discovered a fair number of copies with simple annotations in the opening cosmological chapters, only to have them peter out as the going became more mathematical. In the early chapters Copernicus gave his strongest arguments for a Sun-centered blueprint for the planetary system—arguments based on simplicity, harmony, and aesthetics, for in those days before the invention of the telescope it was impossible to find satisfying observational proofs for the mobility of the Earth. Like Ptolemy's system, Copernicus' heliocentric model gave tolerably good predictions, and many astronomers were fascinated just to see another way to get more or less the same predictions. As the Jesuit astronomer Christopher Clavius at the Vatican's Collegio Romano remarked, Copernicus simply showed that Ptolemy's arrangement was not the only way to do it. But any astronomer who wanted to be sure how it worked in detail had to be well tutored in geometry.
In the decade before I became thoroughly enmeshed with Copernicus, I had already entered the magic circle of Johannes Kepler, the German astronomer who was born almost exactly a century after his Polish forerunner. Kepler's Quadricentennial came in 1971, just two years before the Copernican Quinquecentennial. I sometimes say, with considerable truth, that my career in astrophysics was derailed by anniversaries.
Johannes Kepler burning the
midnight oil as he contemplated the
dome required to finish the Temple
of Urania, from the frontispiece of
his Rudolphine Tables
(Ulm, 1627).
Kepler, the man who really forged the Copernican system as we know it, had been the hero of Arthur Koestler's The Sleepwalkers. Some critics said his book should have been entitled The Sleepwalker, because he had only one good example of a scientist groping in the dark, Kepler himself. A few even more perceptive critics said he had zero examples, an opinion borne out today by recent scholarship. When Kepler wrote in his Astrono-mia nova about finding the elliptical shape of Mars's orbit, saying, "it was as if I had awakened from sleep," readers had little reason to suspect that the account was anything other than straightforward autobiography. They had no way of knowing that it was actually a highly structured rewriting of his personal history of discovery, designed to persuade his readers to abandon the perfect orbital circles, an idea almost as radical as heliocentrism had been a few generations earlier. But Kepler was not sleepwalking at all; the manuscripts show that he knew what he was doing, and even seemingly blind alleys brought grist to his mill. Kepler knew that he had access, through his mentor, Tycho Brahe, to observations far more accurate than any available to Ptolemy or Copernicus, and these showed that the earlier computational methods were by no means adequate, particularly for the planet Mars. New wine required new bottles.
For all its faults, Koestler's book was, for me, a fascinating read, and highly stimulating. When I read it in 1959,1 was about to become heavily involved in astrophysical computing, exploiting the newly available power of the electronic computer revolution. Several years later, in looking around for a historical computing problem as sort of a busman's holiday, hopefully with a Keplerian connection, I encountered an intriguing statement in the early part of the Astronomia nova, where Kepler wrote, "Dear reader, if you are tired by this tedious procedure, take pity on me, for I carried it out at least 70 times." Kepler was trying to find an eccentric circular orbit for Mars based on four observations, and since he could not find a direct answer, he sought the answer through a series of iterations. Computers are particularly good at repetitive problems, and this seemed like a perfect kind of demonstration. So I programmed Kepler's geometry for the Smithsonian Observatory's IBM 7094, fed in his observational data, and the computer solved the problem in eight seconds with nine tries, the minimum required. Today's computers would solve it in a split second, but in 1964 eight seconds seemed like lightning, and the computer magazines just loved this result.
But I was left with a disturbing puzzle. If the IBM 7094 could solve the problem with the minimum of nine tries, why did it take Kepler at least seventy attempts? Did he make so many nume
rical errors that the iterations simply failed to converge? One way to find out, I knew, would be to examine the manuscript record. Unlike the situation with respect to Copernicus, where, apart from the manuscript of De revolutionibus itself, very few manuscript research materials survive, for Kepler there is a huge and only partially mined archive. For the most part, Kepler's manuscript legacy is today found in St. Petersburg, Russia. A modest amount of library research disclosed that the relevant pages might be found in volume 14 of the Kepler papers in the Academy of Sciences Archives in what was then still Leningrad. Thus, in 1965 I requested a microfilm of the manuscripts in that volume. Once every six months for five years I repeated my request to the Soviet authorities and to the Russian astronomers I had met at international astronomy meetings. Finally, in 1970, I actually got the microfilm, and a superb film it was. The archivists had disbound the volume so that it was possible to photograph the complete pages, with nothing lost in the central gutter of the binding.
Armed with the microfilm, I soon learned a lesson that I've been obliged to relearn several times. One can, in the absence of evidence, reconstruct rationally with clear hindsight how a discovery might have taken place. But Kepler was treading where no investigator had gone before, toiling with the murky, ambiguous realities of cutting-edge science, and what really happened was quite distinct from a tidy rational reconstruction. It is true that Kepler was prone to make numerical errors, but this was not his basic problem. He had, at that point in his researches, gained access to the groundbreaking observations of the great and noble Danish observer Ty-cho Brahe. Kepler could see that neither Ptolemy nor Copernicus was able to predict positions to a high degree of accuracy. Compared to anything available to his predecessors, Brahe's legacy was overwhelming. Nevertheless, Kepler could rarely find precisely the right observations he needed, so he was obliged to interpolate from other observations made around the same time. This process itself led to errors, and Kepler had to carry out multiple iterations just to find out where the discrepancies arose. Examining this process, and looking more broadly at the manuscript material, enabled me to warn my colleagues, during the quadricentennial proceedings in 1971, that, contrary to the received opinion, Kepler's Astronomia nova was far from being a simple, linear, autobiographical account of how he had arrived at his conclusions about the planet Mars.
In those years leading up to the time when I received the Leningrad microfilm, I became increasing intrigued by the technical contents of Kepler's Astronomia nova, which ranks alongside Copernicus' De revolutionibus and Newton's Principia in the trilogy of foundation works for the astronomical revolution of the sixteenth and seventeenth centuries. Remarkably enough, unlike the Revolutions and the Mathematical Principles of Natural Philosophy, Kepler's New Astronomy had never been translated into English. Aided by two well-trained classics students, I resolved to remedy this hiatus. And that is how I encountered Kepler's Lenten pretzel.
Kepler's Lenten pretzel for the geocentric path of Mars, from chapter 1 of his Astronomia nova (Prague, 1609).
Kepler's Astronomia nova was subtitled Commentary on the Planet Mars; he opened his commentary with the observational problem he would attempt to solve, namely, to account for the appearances of Mars as seen from the Earth, and this he illustrated with a carefully made diagram showing how Mars tracked with respect to the Earth between the years 1580 and 1596. His diagram, shown above, was an astronaut's-eye view as seen from far above the Earth and above the plane of Mars's orbit. Mars repeatedly approaches the Earth (shown fixed in the center at point a), makes a backward loop, only to recede and repeat the process roughly two years later at a spot in the zodiac about fifty degrees to the east of the previous loop. The loops themselves trace around the entire sky in approximately seventeen years, during which time Mars itself circumnavigates the sky eight times.
In his Latin text Kepler said that he was first inclined to liken his diagram to a ball of yarn, but he thought the better of that and preferred instead to call it a panis quadragesimalis. I recognized panis as meaning "bread," but what to make of quadragesimalis? A fortieth? Forty times? The word isn't in any Latin dictionary, but I remembered some advice given to me by a Harvard mentor, Professor I. B. Cohen, who suggested that when you are stuck on a technical Renaissance Latin word, try the second edition of Webster's unabridged dictionary (which, unlike the third edition, still includes many obsolete words) or the Oxford English Dictionary. And there it was: "belonging to the period of Lent; Lenten." This, in turn, led to an investigation into the history of pretzels.
At that time my wife, Miriam, and I had acquired an Encyclopaedia Britannica, and part of the salesman's pitch was that if the set did not include an answer to any reasonable question, their research team would investigate. I felt an obligation to send them questions with some regularity, as I assumed this would give employment to impoverished University of Chicago graduate students. The only time I felt fully satisfied by an answer was when I inquired about the history of pretzels. The Britannica's investigator reported that pretzels had their origin in southern Germany— Kepler territory—as Lenten favors for children.
Kepler used his Lenten pretzel diagram as the starting point to show how various cosmological models accounted for this convoluted geocentric pattern. It was Claudius Ptolemy, working in Hellenistic Alexandria around the year A.D. 150, who first showed that a relatively simple geometric model could account for the seemingly complex movements of Mars and of the other planets. As the Eames machine showed, he accomplished this with two circles, a smaller planet-bearing circle that rode upon a larger deferent circle.
A careful inspection of Kepler's complex Lenten pretzel reveals that the loops differ from one another not only in how close they come to the Earth but in their width and in their spacing. Ptolemy added two more features to his model to take these aspects into account. First, he moved the center of the deferent circle away from the Earth, to the position marked b in Kepler's pretzel—that could account for the fact that on one side the loops don't come as close as on the other side. This off-center position of the deferent circle gave it an alternative name: an eccentric circle. Ptolemy could not see the loops from above, however, so he had to deduce that this was happening just from the projection of these effects onto the sky.
Second, he had to figure out a way to make the epicycle move around the eccentric (deferent) circle more slowly on the side where the loops didn't come as close to the Earth, and here he invented a very ingenious device called the equant. The equant point E is shown in the diagram below. The angular motion is uniform about that point, so that the epicycle moves from A to B in the same time that it takes to go from C to D, because the angles at E are identical. Of course the epicycle had to travel faster in the segment CD than in AB because the length of the arc is greater.
The equant got Ptolemy into a lot of trouble as far as many of his successors were concerned. It wasn't that his model didn't predict the angular positions satisfactorily. Rather, the equant forced the epicycle to move nonuniformly around the deferent circle, and that was somehow seen as a deviation from the pure principle of uniform circular motion. Ptolemy himself was apologetic about it, but he used it because it generated the motion that was observed in the heavens. Altogether his system was admirably simple considering the apparent complexity and variety of the retrograde loops.
THE MOTTO THAT Erasmus Reinhold wrote on the title page of his De revolutionibus, "Celestial motion is circular and uniform, or composed of circular and uniform parts," was clearly a potshot at Ptolemy and an accolade for Copernicus. Today we admire Copernicus for having the audacity to introduce the heliocentric cosmology into Western culture, essentially triggering the Scientific Revolution. The Copernican cosmology did not just provide the modern blueprint for the solar system. It was a compelling unification of the disparate elements of the heavenly spheres. The greatest of scientists have been unifiers, men who found connections that had never before been perceived. Isaac Newton destroyed th
e dichotomy between celestial and terrestrial motions, forging a common set of laws that applied to the Earth and sky alike. James Clerk Maxwell connected electricity and magnetism, and showed that light was electromagnetic radiation. Charles Darwin envisioned how all living organisms were related through common descent. Albert Einstein tore asunder the separation between matter and energy, linking them through his famous E = mc2 equation.
Copernicus, too, was nothing if not a unifier. In the Ptolemaic astronomy each planet was a separate entity. True, they could be stacked one after another, producing a system of sorts, but their motions were each independent. The result, Copernicus wrote, was like a monster composed of spare parts, a head from here, the feet from there, the arms from yet another creature. Each planet in Ptolemy's system had a main circle and a subsidiary circle, the so-called epicycle. Mars with its epicycle was a prototype for each of the other planets, but because the frequency and size of the retrograde was different for each planet, an epicycle with an individual size and period was required for each planet. Copernicus discovered that he could eliminate one circle from each set by combining them all into a unified system. Just as the Eames machine demonstrated that a heliocentric orbit for the Earth and for Mars could give the same results as a geocentric Martian deferent and epicycle, the same could be done for each planet. If all the models could be scaled so that the Earth's orbit was always the same size, then they could all be stacked together with a single Earth-orbit, thereby reducing the total number of circles. And when Copernicus did this, something almost magical happened. Mercury, the swiftest planet, circled closer to the Sun than any other planet. Lethargic Saturn, then the most distant planet yet identified, circled farthest from the Sun, and the other planets fell into place in between, arranged in distance by their periods of revolution.
The Book Nobody Read Page 6