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 numerical 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 the 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.
"In no other way do we find a wonderful commensurability and a sure harmonious connection between the size of the orbit and the planet's period," Copernicus declared in the most soaring cosmological passage in his entire book. What Copernicus had achieved was a linked system in which all the distances were locked into place relative to a common measure, the Earth-Sun distance, which provided the yardstick for the entire system.* But Reinhold and his many followers admired Copernicus for a quite different aesthetic idea, the elimination of the equant. Copernicus devoted the great majority of De revolutionibus to showing how this could be done. While he had eliminated all of Ptolemy's major epicycles, merging them all into the Earth's orbit, he then introduced a series of little epicyclets to replace the equant, one per planet.* Because this made the motion uniform in each Copernican circle, the anti-equant aesthetic was satisfied. My Copernican census eventually helped to establish that the majority of sixteenth-century astronomers thought eliminating the equant was Copernicus' big achievement, because it satisfied the ancient aesthetic principle that eternal celestial motions should be uniform and circular or compounded of uniform and circular parts.
JOHANN STOEFFLER'S Ephemeridum opus of 1532 was one of the first rare books I had bought for my own library. I found it on the shelf of Blackwell's rare book department in Oxford. In the days when a typical scholarly book cost maybe $510, it was quite a plunge to spend 3170 for a book of numbers, but it was exciting to have such an old volume for my very own. Before that I had used a similar volume in Harvard's Houghton Library to help date a poem by the satirical Tudor poet John Skelton, so I had an idea about what I was getting.† Stoeffler's ephemerides were filled with daily positions of the Sun and planets for the years 1532 through 1551.
I was particularly curious about the basis for Stoeffler's numbers because of a popular legend found in many secondary sources. According to that story, a principal reason why Copernicus had sought to create a heliocentric cosmology was that Ptolemy's hoary system had become so encumbered with jury-rigged embroideries that it was at the point of collapse. Astronomers throughout the ages had added one epicycle to another as they attempted to keep up with the observed deficiencies of the system.
Probably the legend had begun soon after the modern recovery around 1880 of Copernicus' Commentariolus, or "Little Commentary." After describing the complexities of planetary motion, Copernicus closed this account with an exclamation: "Behold! Only 34 circles are required to explain the entire structure of the universe and the dance of the planets!" Superficially, the passage looks as if Copernicus were crowing about the great simplification his system afforded. If Copernicus could handle this in only 34 circles, Ptolemy (or at least his medieval successors) must have required many more.
There is a wonderful, very old, but no doubt apocryphal, story that Alfonso the Great, looking over the shoulders of his astronomers who were compiling the Alfonsine Tables, remarked that if he had been around at Creation, he could have given the Good Lord some hints. The obvious interpretation was that King Alfonso's astronomers, in order to take care of the observed discrepancies between the Ptolemaic predictions and where the planets actually were, had been obliged to add more circles, small epicycles on epicycles. It's rather reminiscent of the lines paraphrasing Jonathan Swift:
Great fleas have little fleas
upon their backs to bite 'em
And little fleas have lesser fleas
and so ad infinitum.
The legend reached its apotheosis when the 1969 Encyclopaedia Britannica announced that, by the time of King Alfonso, each planet required 40 to 60 epicycles! The article concluded, "After surviving more than a millennium, the Ptolemaic system failed; its geometrical clockwork had become unbelievably cumbersome and without satisfactory improvements in its effectiveness." When I challenged them, the Britannica editors replied lamely that the author of the article was no longer living, and they hadn't the faintest idea if or where any evidence for the epicycles on epicycles could be found.
Johann Stoeffler from his Ephemeridum opus (Tubingen, 1531). At least a century ago this image became confused as a portrait of Copernicus.
In those early years of the space age, the Smithsonian Observatory's computer spent most of its time tracking satellites. In its spare time, it calculated the flow of photons through the outer layers of stars—that was my specialty—and in my own spare time I had the machine calculate medieval planetary tables. I recomputed the Alfonsine Tables and discovered to my surprise that they were pure Ptolemaic, totally lacking any embroideries at all. Then, using the hundreds of cards the keypunchers had produced for me, I generated a section of the Stoeffler ephemerides. Again a surprise! My pure and simple Alfonsine Tables calculations closely matched the positions that the Tubingen astronomer had published in his book. His were the best ephemerides of the day, and they showed absolutely no evidence of epicycles on epicycles. A much-repeated and well-entrenched myth had just bit the dust, or so I thought.
During 1973, the great quinquecentennial year, I had occasion to mention my conclusions at a symposium in Copernicus' birthplace, Toruh. In the audience was Edward Rosen, the world's foremost Copernican scholar. A professor at City College in New York, he had built his career of fastidious scholarship on finely honed translations and the ability to dig out relevant details from incredibly obscure sources. As part of his doctoral dissertation, Rosen had translated Copernicus' Commentariolus, and he was particularly fond of the line about the entire ballet of the planets being accomplished in just 34 circles; he firmly believed that part of Copernicus' achievement was to simplify an overwrought system. "How can you be so sure there weren't epicycles on epicycles?" he demanded to know. Certainly, I hadn't inspected all the possible medieval manuscripts!
I'm not sure I ever convinced him about epicycles on epicycles, but today I understand the problem a lot better. The entire calculational procedure for the Alfonsine Tables depends on a clever approximation invented by Ptolemy to handle a single epicycle on an eccentric circle. Frankly, there was no mathematician in the Middle Ages ingenious enough to have devised a similarly economical computational scheme for multiple epicycles. It's not even necessary to inspect all the medieval astronomical manuscripts to be sure.
Edward Rosen had simply misread Copernicus' intentions in writing that the entire ballet of the planets was accomplished in only 34 circles. Copernicus must have realized that with his small epicyclets he actually had more circles than the Ptolemaic computational scheme used in the Alfonsine Tables or for the Stoeffler ephemerides. His exuberant conclusion to the Commentariolus surely registered his delight that, although the celestial appearances seem very complicated, a great many phenomena can be modeled with only 34 circles. There was clearly no comparison intended with his predecessors.
THE EPICYCLES-ON-EPICYCLES legend fails in yet another way. There are virtually no records of systematic observations to find possible discrepancies between where the tables predicted the planets to be and where they really were. Yet there is one minor but extraordinarily significant exception that I discovered when Charles Eames and I were photographing the Copernican books in Uppsala.
Bound in the back of his printed copy of the Alfonsine Tables are sixteen extra leaves on which Copernicus added carefully written tables and miscellaneous notes. Below the record of two observations made in Bologna in 1500, there is, in another ink, a cryptic undated remark in abbreviated Latin: "Mars surpasses the numbers by more than two degrees. Saturn is surpassed by the numbers by one and a half degrees" (plate 7a)."
In 1504, soon after Copernicus had returned to Poland from Italy with his newly acquired doctorate, the two slowest-moving naked-eye planets put on a splendid show as faster-moving Jupiter bypassed the slower-moving Saturn. This great conjunction, which takes place only every twenty years, provided a sensitive test of the tables, since it did not require any elaborate instruments to determine on which night the planets actually passed each other. And this time Mars joined the ballet. Between October 1503 and March 1504, swifter Mars passed Jupiter and then Saturn, and then, going into retrograde motion, went back past Saturn and Jupiter, and finally in direct motion bypassed Jupiter and Saturn yet again. It was a great celestial display, and surely Copernicus would not have missed it.
With my computer programs I could calculate where the Alfonsine Tables placed these planets not just during this period but for several decades of the sixteenth century, and I could compare these calculations with modern ones showing where the planets really were. To my amazement, the calculations gave a unique error signature for February and March in 1504. During that interval the predictions for Jupiter were excellent, but Saturn lagged behind the tables by a degree and a half while Mars went ahead of the predictions by nearly two degrees. Only during this interval did the errors match Copernicus' notes, so the evidence is firm that he had observed the cosmic dance at this time and was fully aware of the discrepancies in the tables. But what is most astonishing is that Copernicus never mentioned his observation, and his own tables made no improvement in tracking these conjunctions.
It is pretty clear that neither Copernicus nor his predecessors were interested in adding extra circles just to make the predictions work a little better. Nevertheless, the legend of epicycles on epicycles has become so pervasive that barely a year passes without some author in the Physical Review or the AstronomicalJournal remarking, apologetically, "Maybe my theory has too many epicycles." Clearly, I haven't stamped out the myth.
* Surprisingly, Copernicus' planetary system was more compact than the carefully nested pieces of the Ptolemaic arrangement. Yet his cosmos was vastly larger than Ptolemy's because he was obliged to place the stars themselves far enough away so that the motion of the Earth around the Sun would not show any obvious changes in the positions of the stars. "So vast, without any question, is the divine handwork of the Almighty Creator," Copernicus concluded. It was a giant step for humankind, in the right direction, but ultimately that sentence made the Catholic censors very nervous. Probably, they had no objections to the vastness, but they just weren'r convinced that Copernicus could know how God did it. Catholics were ordered to delete that sentence from their copies.
* For the mathematically curious, details of Copernicus' procedure are given in appendix 1. In the Commentariolus Copernicus used a double epicyclet for Venus, Mars, Jupiter, and Saturn, but in De revolutionibus only a single epicyclet with an eccentric orbit. The Earth and Mercury had somewhat more complicated configurations—it remained for Kepler to construct a truly unified Copernican system.
† Skelton's "Garland of Laurel" included the lines
Arectyng my syght to the Zodyake, . . .
When Mars rerrogradanr reversyd his bak . . .
And whan Lucina plenarly did shyne,
Scorpione ascending degrees twyse nyne.
Using the Regiomontanus Ephemerides for 1495,1 could see that on 8 May 1495, the Moon was full in the eighteenth degree of Scorpio and in conjunction with the retrograding Mars.