The Hunt for Vulcan

by Thomas Levenson

For Newton the comet of 1680 offered a unique opportunity. He already knew the shapes of the planetary orbits he analyzed with his new mathematical laws—but this previously unknown visitor presented a novel challenge: could his universal gravitation account for motion no one had seen before? He set up his analysis by first plotting the path of Kirch’s comet as revealed in reports from credible observers. He drew a line that connected each observation to reveal its track: a particular kind of curve called a parabola. Parabolas are mathematically kin to the ellipses traced out by the planets and moons Newton had just analyzed. The key difference: ellipses are closed curves; the earth, the planets, Halley’s comet, NASCAR drivers*2 retrace their path with every trip round their oval courses. Not so any object on a parabolic path. Parabolas are open-ended, following lines that start out there, bend round a focus (the sun, for the comet of 1680), and shoot off again on a course that will never return to the old neighborhood.

  Newton made sure every reader really, really understood that yes, the comet of 1680 rode a parabola in and out of the solar system. At the end of a very long and difficult book, he devoted page after page to detailed lists of observations from all those comet hunters who had chased it through the constellations. He left nothing out—it was as if he wanted to cudgel his readers into silent agreement. By the end of his account no one could possibly doubt: the comet of 1680 roared in from who-knew-where, rounded the sun…and then vanished into the unmapped vastness beyond, never, apparently, to return.

  And then he performed one last feat. He extracted just three observations from his catalogue, three points on the comet’s trajectory, and used his new mathematical model of force and motion to derive the orbit of that comet. He calculated, and the answer came back a perfect match: his results graphed onto that same course all those observers had found: a parabola.*3 Strip away the technical complexity—all those conics and curves and calculus masquerading as geometry—and what remained was the triumph, not just Newton’s, but that of a whole new way of grasping the material world.

  The account of the comet of 1680 gives his book its true climax. It was cosmic proof that the same laws that governed ordinary experience—the apple’s fall, an arrow’s flight, the moon’s constant path—ruled all experience, to the limits of the universe. A parabola has no end nor beginning: One arm comes from the infinitude of the plane; the other arm shoots off to the same infinity. Placed in the material world, formed by the motion of a comet swinging around the sun, the parabolic motion of the comet of 1680 traces out not just the events that take place in our immediate vicinity, but throughout the universe, from its deepest reaches and back out to them again.

  Newton knew exactly what he had done. Near the end of the section on comets, he wrote: “The theory that corresponds exactly to so nonuniform a motion through the greatest part of the heavens, and that observes the same laws as the theory of the planets, and that agrees exactly with exact astronomical observations cannot fail to be true.”

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  Edmond Halley agreed. Three years after he’d innocently asked for a single proof, he delivered to the printer the last pages of what Newton again immodestly, again accurately, titled Philosophiae Naturalis Principia Mathematica—The Mathematical Principles of Natural Philosophy. Getting Newton’s enormous manuscript into book form while dealing with its ever-fractious author had left no time for Halley’s own work since 1684, but now, at the finishing line, he granted himself his own victory lap. As Principia went to press, he exercised his editor’s privilege to preface Newton’s prose with a poetic assessment of the great work and its author: “But we are now admitted to the banquets of the gods/We may deal with the laws of heaven above; and we now have/The secret key to unlock the obscure earth; and we know the immovable order of the world/…Join me in singing the praises of Newton, who reveals all this,/Who opens the chest of hidden truth.”

  Hidden truths made plain! That was no poetic license. Amid all the talk of gods and heaven, Halley got it right. Newton had promised his readers the system of the world—and this is in fact what they received, a way to investigate matter in motion throughout the cosmos, to the utter limit of space and time. As the great French mathematician Joseph-Louis Lagrange famously said, “Newton was the greatest genius who ever lived, and the most fortunate; for we cannot find more than once a system of the world to establish.”

  Title page from the first edition of Principia

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  Sir Isaac Newton died in 1727. Alexander Pope responded with his famous epigram: “Nature and nature’s ways lay hid in night./God said, ‘Let Newton be,’ and all was light.” By the turn of the next century such apparent hyperbole would seem no more than predictable British understatement.

  * * *

  *1 That object follows its own elliptical path—a much more elongated version of the orbits taken by the major planets—and it completes a single journey around the sun about every seventy-six years. Halley would later crack its true nature as a repeat visitor by analyzing historical sightings within Newton’s gravitational mathematics in one of the early triumphs of the new science.

  *2 For the record: the closed curved tracks on which NASCAR races are run generally are not perfect ellipses.

  *3 Newton later returned to the question of the true orbit of the comet of 1680, considering the possibility that it followed not a parabolic orbit, but instead a very elongated, long-period elliptical one. He was never able to derive such an orbit with confidence, though he believed it could have been a returning comet with a 575-year orbit. More recent analyses put the possible period on the order of ten thousand years.

  “A HAPPY THOUGHT”

  March 1781, Bath.

  It was William Herschel’s day job that brought him to the city of Bath. He was a Hanoverian transplant, a musician by trade who had been made director of the Bath Orchestra in 1780. But if music paid the bills, the stars were his passion, ever since he saw the same view that has grabbed so many amateurs before and since: the rings of Saturn in all their glory.

  Inspired, Herschel had taught himself how to construct a telescope (helped by his sister Caroline, reportedly the better of the two at the fine figuring of mirrors) and he had made the switch from stargazing to systematic astronomy as early as 1774. In Bath he settled into a seemingly prosaic program, an analysis of double stars. His goal: to distinguish these paired flecks of lights as either genuine examples of two stars in close proximity or as “opticals,” two completely unrelated objects that just happened to fall together on a line of sight.

  And so, on Tuesday, March 13, 1781, around the time women of the upper classes would have risen from their dinner table to allow their men to sit over their cigars and drink, he set himself to what had become a regular routine. He turned his largest and newest telescope—a Newtonian design with a 6.2-inch mirror, the best in England—toward a candidate double between the constellations Taurus and Gemini. One half of the apparent pair was utterly undistinguished, an ordinary point of light, just a star. The other? It looked odd: fuzzy. Most important, it changed under magnification. Herschel recorded the anomaly in his entry for the night: “The lower of the two is a curious either nebulous star or perhaps a comet.”

  Credit 2.1

  Thomas Digges’s diagram of the Copernican cosmos, first published in 1576, depicted all the elements of the universe as it was known in the spring of 1781.

  Herschel returned to observe the object repeatedly over the next month, until he was sure that it was indeed a candidate comet, a nearby object moving against a starry background. But by then it was clear that this “comet” was behaving oddly: it did not appear to be growing in the sky (or not much—Herschel persuaded himself for a while that he had measured an increase in its diameter) and it showed no signs of growing a tail. He reported his finding to the Royal Society, and other observers began to examine the object.

  In May, two mathematicians—one French, one Russian—independently used the accumulating sightings to work out its true
orbit. They proved (as Herschel had not) that this wayfarer was no comet. Rather, it traveled a nearly circular orbit, farther from the sun than that astronomer’s gateway drug, the ringed giant Saturn.

  From the beginnings of recorded history to that night in Bath, humankind had known exactly how many wandering stars—planets—traversed the skies. There were just six: Mercury, nearest the sun, then Venus, then our own Earth, Mars, Jupiter, and, most distant, Saturn. Even after 1609 when Galileo turned his strange new device, a tube with disks of figured glass at either end, and added Jupiter’s moons to the solar system’s family tree, that tally remained unchallenged. No longer. By convention, historians of astronomy date the discovery of the planet Uranus to Herschel’s first glimpse on March 13, 1781.

  Unsurprisingly, such an unprecedented find turned Herschel into a hero to his contemporaries. King George III offered him a £200 stipend if he would bring his observatory to Windsor Castle, then added a knighthood to sweeten the pot. Herschel’s fellow astronomers took their reward too. Uranus created a unique opportunity, as it was the first major finding that could pose an independent test of Newton’s mathematical version of reality. Put another way: a previously unknown object offered the astronomical community a chance to see how well their fundamental tools actually accommodated not just what was known, but what had, until that March evening, remained unsuspected.

  Credit 2.2

  Pierre-Simon Laplace, in a posthumous portrait by Sophie Feytaud

  Among the first to take up the challenge was a young, brilliant French mathematician: Pierre-Simon Laplace. Laplace was something of a prodigy. Elected to the Académie Royale des Sciences eight years earlier, just twenty-four, he had since presented leading-edge work on pure mathematics, gravitation, probability, and more. When he heard the news of Herschel’s observations, he immediately joined what was almost a stampede of European thinkers applying Newtonian ideas to the still unidentified object. Like Herschel himself, Laplace leapt to the conclusion that the object was a comet (which, of course, was hardly a crazy assumption: plenty of comets had been seen before and after the beginnings of telescopic astronomy, but no one had turned up any new planets—until then).

  His attempt to calculate a plausible cometary trajectory failed, though, and he left it to others to expose Uranus’s true nature. After that, though, Laplace took up the data again, and by early 1783 had come up with a novel, more general method for analyzing the motion of celestial objects. When he applied his new approach to Uranus, he produced the best description of its orbit yet achieved. To Laplace, that calculation was at once a rather minor display of analytical skill and one of the opening shots in what would become his life’s work: using Newtonian physics, expressed in ever more sophisticated mathematics, to complete old Sir Isaac’s foundational program, the construction in detail of a “system of the world” that could account for the behavior of every object in the universe, those known, and those yet to be discovered.

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  That work occupied Laplace for the next three decades and more. Between the 1780s and the first years of the new century, he created the most comprehensive account anyone had yet achieved of the interactions of the sun, its planets, their moons. With the rigor imposed by his increasingly sophisticated mathematical language, Laplace transformed the apparatus Newton used to show that the universe could be made intelligible into an epic narrative of how the universe actually behaves.

  It wasn’t always clear that the work would reach that happy ending. By the late eighteenth century, the dynamics of the solar system faced some open questions—several unanswered for decades. The most pressing: Jupiter was moving faster by the end of the seventeenth century than it had appeared to travel in earlier records, while Saturn seemed to have slowed down. The simplest analysis of the system (the sort Newton himself performed in Principia) implied that this couldn’t happen. But manifestly it did, as documented by none other than Newton’s dear friend Halley.

  Enter Laplace, with a virtuoso demonstration of how his version of postrevolutionary science created new knowledge. Newton’s gravitation boils down to this: an equation that tells you exactly how two bodies influence each other. If you know a handful of basic parameters—the masses of two bodies influencing each other, the distance between them—you can just turn the crank to figure out how much force each imposes on the other.* Going from that force to a trajectory, an orbit, or the flight of a comet (or a cannonball) is a bit more complicated, but not much.

  But such calculations are always idealized. Most of the time reality is messy, and the simplest applications of fundamental laws don’t survive much contact with the world. The true test of Newton’s science—of any abstract claim—comes when there is a conflict between existing understanding and some fact that doesn’t fit. The failure to match the actual motions of Saturn and Jupiter with what the theory seemed to say should happen posed the question: what does that conflict mean? Is that a problem, or an opportunity?

  Laplace held to his credo: “there is no truth in physics,” he wrote, “more indisputable and better established by the agreement between observation and calculation than that all celestial bodies gravitate toward each other.” This was, he added, Newton’s doing, the outcome of the “most important discovery ever made in natural philosophy.” The key to Laplace’s adulation, though, lay with the necessity that observation and calculation agree on Newton’s discovery. What to do, then, were they to disagree? As Laplace certainly knew, when the real world confounded theoretical explanation, that could simply mean that the theory was wrong. But there was another option. If something that can be measured doesn’t fit, Laplace reasoned, the obvious next step is to look for something else, some other fact, perhaps some new way to understand the math itself, that could haul the real world back into agreement with its mathematical representation. Put another way: something out of whack suggests that there is something else out there to discover, maybe in nature, perhaps within the abstract ideas built to interpret nature’s ways.

  Laplace set to work on Jupiter and Saturn in 1785. He began on solid ground. The plain reading of Newton’s laws said that Saturn and Jupiter should interact, and that the results of their gravitational dance could indeed be the kind of motion actually observed, the larger planet accelerating and the smaller slowing down. He reworked the calculation others had attempted before him and came to the same answer: the scale of the observed braking and speeding was about right, with a deviation small enough to support the intuition that the source of the error lay not with any failure by Newton, but in something Newton’s heirs had missed.

  With his own doubts thus relieved, Laplace next attempted what no one had yet been able to do: to construct a mathematical approach that would treat Jupiter and Saturn as a continuously changing series of related systems. Each change in the relative position of the two planets implied a different set of inputs for the equations that translate gravity into motion for each object. The “error”—that little bit of unaccountable extra acceleration of Jupiter—would, if this approach worked out, reveal itself as a perfectly “natural” outcome of the mathematics of gravity describing patterns of motion that evolve over time. This was a masterful bit of intellectual jujitsu, a shift from the observed behavior of celestial objects to the mental picture his fellow mathematical philosophers had constructed to model that behavior.

  A map of the solar system published in 1791 as part of the Tom Telescope series of science books for children. In this very British setting, Uranus is still known as “the Georgian Planet”—an attempt at interplanetary nationalism that didn’t last long.

  There was only one difficulty. Representing the two planets’ positions and their relations to each other in three spatial dimensions and then allowing time to flow forced Laplace to construct a devilishly complicated system of equations. That mathematical model could only be solved (maybe) by an equally intimidating set of calculations. In the end, it took him a full three years—and that only with some ex
tremely skilled help on the necessary analytical grunt work. But finally, in 1788, he was able to announce that he had cracked the mystery. The observed acceleration of Jupiter and deceleration of Saturn, he said, were caused by minute changes in the strength of the gravitational attraction between the two as their trajectories shifted. These changes occurred on a cycle that spanned centuries—929 years, in fact. The test of this claim turned on checking the exact timing of particular orbital conformations as far back as records could reveal—and in a test of theory against observation since 228 B.C.E., Laplace was able to show that the two planets obeyed Newtonian theory to the limits of the accuracy of available measurements.

  It was a bravura performance, an almost ridiculously accomplished display of mathematical skill. It achieved something more too, a confirmation not just of Newton’s theory as Laplace’s “indisputable truth”—but of the truly revolutionary nature of the scientific revolution itself. Laplace had developed mathematical tools through which he expanded the reach of the fundamental laws Newton himself had created. This new math permitted a sharper description—more resolution in the picture, as it were—of the physical behavior it sought to model. Most important: that image wasn’t simply more precise. It contained novel information, more detail; in this case, a hidden, slow dance of planets that takes nearly a millennium to unfold.

  Thus the deep power of Newtonian science as Laplace and his peers understood it: it was an engine of discovery, powered by reason expressed in the particular rigor of mathematics. There was no end to the usual forms of exploration—the recent addition of Uranus to the solar system, for example, demonstrated that each advance in the technology of scientific instruments could reveal unsuspected terrain. But as Newton’s followers and interpreters drove ever deeper into his mathematical reimagining of natural philosophy, it became clear that it was possible to explore within the math as well, journeys of the mind that could leap the page and guide explorers to discoveries in the wide world itself.