CHAPTER XIX

THE SEARCH FOR A TRANS-NEPTUNIAN PLANET

We must now return to the last paragraph of his “Memoir on the Origin of the Planets,” where he suggests the probable distance of a body beyond Neptune. In fact he had long been interested in its existence and whereabouts. By 1905 his calculations had given him so much encouragement that the Observatory began to search for the outer planet, which he then expected would be like Neptune, low in density, large and bright, and therefore much more easily detected than it turned out to be. But the photographs taken in 1906, with a well planned routine search the next year revealed nothing, and he became distrustful of the data on which he was working. In March 1908, one finds in his letter-books from the office in Boston the first of a series of letters to Mr. William T. Garrigan of the Naval Observatory and Nautical Almanack about the residuals of Uranus—that is the residue in the perturbations of its normal orbit not accounted for by those due to the known planets. He suggests including later data than had hitherto been done; asks what elements other astronomers had taken into account in estimating the residuals; points out that for different periods they are made up on different theories in the publications of Greenwich Observatory, and that some curious facts appear from them. About his own calculation he writes on December 28, 1908: “The results so far are both interesting and promising.” He was hard at work on the calculations for such a planet, based upon the residuals of Uranus, and assisted by a corps of computers, with Miss Elizabeth Williams, now Mrs. George Hall Hamilton of the Observatory at Mandeville, Jamaica, at their head.

Before trying to explain the process by which he reached his results it may be well to give his own account of the discovery of Neptune by a similar method:[39]

“Neptune has proved a planet of surprises. Though its orbital revolution is performed direct, its rotation apparently takes place backward, in a plane tilted about 35° to its orbital course. Its satellite certainly travels in this retrograde manner. Then its appearance is unexpectedly bright, while its spectrum shows bands which as yet, for the most part, defy explanation, though they state positively the vast amount of its atmosphere and its very peculiar constitution. But first and not least of its surprises was its discovery,—a set of surprises, in fact. For after owing recognition to one of the most brilliant mathematical triumphs, it turned out not to be the planet expected.

“‘Neptune is much nearer the Sun than it ought to be,’ is the authoritative way in which a popular historian puts the intruding planet in its place. For the planet failed to justify theory by not fulfilling Bode’s law, which Leverrier and Adams, in pointing out the disturber of Uranus, assumed ‘as they could do no otherwise.’ Though not strictly correct, as not only did both geometers do otherwise, but neither did otherwise enough, the quotation may serve to bring Bode’s law into court, as it was at the bottom of one of the strangest and most generally misunderstood chapters in celestial mechanics.

“Very soon after Uranus was recognized as a planet, approximate ephemerides of its motion resulted in showing that it had several times previously been recorded as a fixed star. Bode himself discovered the first of these records, one by Mayer in 1756, and Bode and others found another made by Flamsteed in 1690. These observations enabled an elliptic orbit to be calculated which satisfied them all. Subsequently others were detected. Lemonnier discovered that he had himself not discovered it several times, cataloguing it as a fixed star. Flamsteed was spared a like mortification by being dead. For both these observers had recorded it two or more nights running, from which it would seem almost incredible not to have suspected its character from its change of place.

“Sixteen of these pre-discovery observations were found (there are now nineteen known), which with those made upon it since gave a series running back a hundred and thirty years, when Alexis Bouvard prepared his tables of the planet, the best up to that time, published in 1821. In doing so, however, he stated that he had been unable to find any orbit which would satisfy both the new and the old observations. He therefore rejected the old as untrustworthy, forgetting that they had been satisfied thirty years before, and based his tables solely on the new, leaving it to posterity, he said, to decide whether the old observations were faulty or whether some unknown influence had acted on the planet. He had hardly made this invidious distinction against the accuracy of the ancient observers when his own tables began to be out and grew seriously more so, so that within eleven years they quite failed to represent the planet.

“The discrepancies between theory and observation attracted the attention of the astronomic world, and the idea of another planet began to be in the air. The great Bessel was the first to state definitely his conviction in a popular lecture at Königsberg in 1840, and thereupon encouraged his talented assistant Flemming to begin reductions looking to its locating. Unfortunately, in the midst of his labors Flemming died, and shortly after Bessel himself, who had taken up the matter after Flemming’s death.

“Somewhat later Arago, then head of the Paris observatory, who had also been impressed with the existence of such a planet, requested one of his assistants, a remarkable young mathematician named Leverrier, to undertake its investigation. Leverrier, who had already evidenced his marked ability in celestial mechanics, proceeded to grapple with the problem in the most thorough manner. He began by looking into the perturbations of Uranus by Jupiter and Saturn. He started with Bouvard’s work, with the result of finding it very much the reverse of good. The farther he went, the more errors he found, until he was obliged to cast it aside entirely and recompute these perturbations himself. The catalogue of Bouvard’s errors he gave must have been an eye-opener generally, and it speaks for the ability and precision with which Leverrier conducted his investigation that neither Airy, Bessel, nor Adams had detected these errors, with the exception of one term noticed by Bessel and subsequently by Adams.[40] The result of this recalculation of his was to show the more clearly that the irregularities in the motion of Uranus could not be explained except by the existence of another planet exterior to him. He next set himself to locate this body. Influenced by Bode’s law, he began by assuming it to lie at twice Uranus’ distance from the Sun, and, expressing the observed discrepancies in longitude in equations, comprising the perturbations and possible errors in the elements of Uranus, proceeded to solve them. He could get no rational solution. He then gave the distance and the extreme observations a certain elasticity, and by this means was able to find a position for the disturber which sufficiently satisfied the conditions of the problem. Leverrier’s first memoir on the subject was presented to the French Academy on November 10, 1845, that giving the place of the disturbing planet on June 1, 1846. There is no evidence that the slightest search in consequence was made by anybody, with the possible exception of the Naval Observatory at Washington. On August 31 he presented his third paper, giving an orbit, mass, and more precise place for the unknown. Still no search followed. Taking advantage of the acknowledging of a memoir, Leverrier, in September, wrote to Dr. Galle in Berlin asking him to look for the planet. The letter reached Galle on the 23rd, and that very night he found a planet showing a disk just as Leverrier had foretold, and within 55′ of its predicted place.

“The planet had scarcely been found when, on October 1, a letter from Sir John Herschel appeared in the _London Athenaeum_ announcing that a young Cambridge graduate, Mr. J. C. Adams, had been engaged on the same investigation as Leverrier, and with similar results. This was the first public announcement of Mr. Adams’ labors. It then appeared that he had started as early as 1843, and had communicated his results to Airy in October, 1845, a year before. Into the sad set of circumstances which prevented the brilliant young mathematician from reaping the fruit of what might have been his discovery, we need not go. It reflected no credit on any one concerned except Adams, who throughout his life maintained a dignified silence. Suffice it to say that Adams had found a place for the unknown within a few degrees of Leverrier’s; that he had communicated these results to Airy; that Airy had not considered them significant until Leverrier had published an almost identical place; that then Challis, the head of the Cambridge Observatory, had set to work to search for the planet but so routinely that he had actually mapped it several times without finding that he had done so, when word arrived of its discovery by Galle.

“But now came an even more interesting chapter in this whole strange story. Mr. Walker at Washington and Dr. Petersen of Altona independently came to the conclusion from a provisional circular orbit for the newcomer that Lalande had catalogued in the vicinity of its path. They therefore set to work to find out if any Lalande stars were missing. Dr. Petersen compared a chart directly with the heavens to the finding a star absent, which his calculations showed was about where Neptune should have been at the time. Walker found that Lalande could only have swept in the neighborhood of Neptune on the 8th and 10th of May, 1795. By assuming different eccentricities for Neptune’s orbit under two hypotheses for the place of its perihelion, he found a star catalogued on the latter date which sufficiently satisfied his computations. He predicted that on searching the sky this star would be found missing. On the next fine evening Professor Hubbard looked for it, and the star was gone. It had been Neptune.[41]

“This discovery enabled elliptic elements to be computed for it, when the surprising fact appeared that it was not moving in anything approaching the orbit either Leverrier or Adams had assigned. Instead of a mean distance of 36 astronomical units or more, the stranger was only at 30. The result so disconcerted Leverrier that he declared that ‘the small eccentricity which appeared to result from Mr. Walker’s computations would be incompatible with the nature of the perturbations of the planet Herschel,’ as he called Uranus. In other words, he expressly denied that Neptune was his planet. For the newcomer proceeded to follow the path Walker had computed. This was strikingly confirmed by Mauvais’ discovering that Lalande had observed the star on the 8th of May as well as on the 10th, but because the two places did not agree, he had rejected the first observation, and marked the second as doubtful, thus carefully avoiding a discovery that actually knocked at his door.

“Meanwhile Peirce had made a remarkable contribution to the whole subject. In a series of profound papers presented to the American Academy, he went into the matter more generally than either of the discoverers, to the startling conclusion ‘that the planet Neptune is not the planet to which geometrical analysis had directed the telescope, and that its discovery by Galle must be regarded as a happy accident.’[42] He first proved this by showing that Leverrier’s two fundamental propositions,—

“1. That the disturber’s mean distance must be between 35 and 37.9 astronomical units;

“2. That its mean longitude for January 1, 1800, must have been between 243° and 252°,—were incompatible with Neptune. Either alone might be reconciled with the observations, but not both.

“In justification of his assertion that the discovery was a happy accident, he showed that three solutions of the problem Leverrier had set himself were possible, all equally complete and decidedly different from each other, the positions of the supposed planet being 120° apart. Had Leverrier and Adams fallen upon either of the outer two, Neptune would not have been discovered.[43]

“He next showed that at 35.3 astronomical units, an important change takes place in the character of the perturbations because of the commensurability of period of a planet revolving there with that of Uranus. In consequence of which, a planet inside of this limit might equally account for the observed perturbations with the one outside of it supposed by Leverrier. This Neptune actually did. From not considering wide enough limits, Leverrier had found one solution, Neptune fulfilled the other.[44] And Bode’s law was responsible for this. Had Bode’s law not been taken originally as basis for the disturber’s distance, those two great geometers, Leverrier and Adams, might have looked inside.

“This more general solution, as Peirce was careful to state, does not detract from the honor due either to Leverrier or to Adams. Their masterly calculations, the difficulty of which no one who has not had some experience of the subject can appreciate, remain as an imperishable monument to both, as does also Peirce’s to him.”

The facts, that is what was done and written, are of course correct; but the conclusions drawn from them are highly controversial to the present day.

The calculations for finding an unknown planet by the perturbations it causes in the orbit of another are extremely difficult, the more so when the data are small and uncertain. For Percival they were very small because Neptune,—nearest to the unknown body,—had been discovered so short a time that its true orbit, apart from the disturbances therein caused by other planets, was by no means certain. In fact Percival tried to analyze its residuals, but they yielded no rational result. This left only what could be gleaned from Uranus after deducting the perturbations caused by Neptune, and that was small indeed. In 1845, when the calculations were made which revealed that planet, “the outstanding irregularities of Uranus had reached the relatively huge sum of 133″. To-day its residuals do not exceed 4.5″ at any point of its path.”

Then there are uncertainties depending on errors of observation, which may be estimated by the method of least squares of the differences between contemporary observations. Moreover there is the uncertainty that comes from not knowing how much of the observed motion is to be attributed to a normal orbit regulated by the Sun, and how much to the other planets, including the unknown. Its true motion under these influences can be ascertained only by observing it for a long time, and by taking periods sufficiently far apart to distinguish the continuing effects of the known bodies from those that flow from an unknown source. This was the ingenious method devised by Leverrier as a basis for his calculations, and he thereby got his residuals caused by the unknown planet in a form that could be handled.

Finally there was the uncertainty whether the residual perturbations, however accurately determined, were caused by one or more outer bodies. Of this Percival was, of course, well aware, and in fact, in his study of the comets associated with Jupiter he had pointed out that there probably was a planet far beyond the one for which he was now in search. But, as no one has ever been able to devise a formula for the mutual attraction of three bodies, he could calculate only for a single body that would account as nearly as possible for the whole of the residuals.

Thus he knew that his work was an approximation; near enough, he hoped, to lead to the discovery of the unknown.

The various elements in the longitude of a planet’s orbit, that is in the plane of the ecliptic, that are affected by and affect another, are:

a—The length of its major, or longest, axis.

n—Its mean motion, which depends on the distance from the Sun.

ε—The longitude at a given time, that is its place in its orbit.

e—The eccentricity of its orbit, that is how far it is from a circle.

ῶ—The place of its perihelion, that is the position of its nearest approach to the Sun.

(These last two determine the shape of the ellipse, and the direction of its longer axis with respect to that of the other planet.)

m—Its mass.

Now formulas, or series of equations, that express the perturbations caused by one planet in the orbit of another must contain all these elements, because all of them affect the result. But there are too many of them for a direct solution. Therefore Leverrier assumed a distance of the unknown planet from the Sun, and with it the mean motion which is proportional to that distance; worked out from the residuals of Uranus at various dates a series of equations in terms of the place of the unknown in its orbit; and then found what place therein at a given time would give results reducing the residuals to a minimum—that is, would come nearest to accounting for them. In fact, supposing that the unknown planet would be about the distance from the Sun indicated by Bode’s law, the limits within which he assumed trial distances were narrow, and, as it proved, wholly beyond the place where it was found. This method, which in its general outline Percival followed, consisted therefore of a process of trial and error for the distance (with the mean motion) and for the place of X in its orbit (ε). For the other three elements (e, ῶ and m) he used in the various solutions 24 to 37 equations drawn from the residuals of Uranus at different dates, and expressed in terms of ε. He did this in order to have several corroborative calculations, and to discover which of them accorded most closely with the perturbations observed.

We have seen that in 1908-09 Percival was inquiring about the exact residuals of Uranus, and he must have been at work on them soon afterwards, for on December 1, 1910, he writes to Mr. Lampland that Miss Williams, his head computer, and he have been puzzling away over that trans-Neptunian planet, have constructed the curve of perturbations, but find some strange things, looking as if Leverrier’s later theory of Uranus were not exact. This work had been done by Leverrier’s methods “but with extensions in the number and character of the terms calculated in the perturbation in order to render it more complete.” Though uncertain of his results, he asks Mr. Lampland, in April 1911, to look for the planet. But he was by no means himself convinced that his data were accurate, and he computed all over again with the residuals given by Gaillot, which he considered more accurate than Leverrier’s in regard to the masses, and therefore the attractions, of the known planets concerned. Incidentally he remarks at this point in his Memoir,[45] in speaking of works on celestial mechanics, that “after excellent analytical solutions, values of the quantities involved are introduced on the basis apparently of the respect due to age. Nautical Almanacs abet the practice by never publishing, consciously, contemporary values of astronomic constants; thus avoiding committal to doubtful results by the simple expedient of not printing anything not known to be wrong.” His result for X, as he called the planet he was seeking, computed by Gaillot’s residuals, differed from that found in using Leverrier’s figures by some forty degrees to the East, and on July 8 he telegraphs Mr. Lampland to look there.

These telegrams to Mr. Lampland continue at short intervals for a long time with constant revisions and extensions in the calculations; and, as he notes, every new move takes weeks in the doing; but all without finding planet X. Perhaps it was this disappointment that led him to make the even more gigantic calculation printed in the Memoir, where he says: “In the present case, it seemed advisable to pursue the subject in a different way, longer and more laborious than these earlier methods, but also more certain and exact: that by a true least-square method throughout. When this was done, a result substantially differing from the preliminary one was the outcome. It both shifted the minimum and bettered the solution. In consequence, the whole work was done _de novo_ in this more rigorous way, with results which proved its value.”

Then follow many pages of transformations which, as the guide books say of mountain climbing, no one should undertake unless he is sure of his feet and has a perfectly steady head. But anyone can see that, even in the same plane, the aggregate attractions of one planet on another, pulling eventually from all possible relative positions in their respective elliptical orbits with a force inversely as the square of the ever-changing distance, must form a highly complex problem. Nor, when for one of them the distance, velocity, mass, position and shape of orbit are wholly unknown, so that all these things must be represented by symbols, will anyone be surprised if the relations of the two bodies are expressed by lines of these, following one another by regiments over the pages. In fact the Memoir is printed for those who are thoroughly familiar with this kind of solitaire.

For the first trial and error Percival assumed the distance of X from the Sun to be 47.5 planetary units (the distance of the Earth from the Sun being the unit), as that seemed on analogy a probable, though by no means a certain, distance. With this as a basis, and with the actual observations of Uranus brought to the nearest accuracy by the method of least-squares of errors, he finds the eccentricity, the place of the perihelion and the mass of X in terms of its position in its orbit. Then he computes the results for about every ten degrees all the way round the orbit, and finds two positions, almost opposite, near 0° and near 180°, which reduce the residuals to a minimum—that is which most nearly account for the perturbations. Each of these thirty tried positions involved a vast amount of computation, but more still was to come.

Finally, to be sure that he had covered the ground and left no loophole for X to escape, he tried, beside the 47.5 he had already used, a series of other possible distances from the Sun,—40.5, 42.5, 45, 51.25 units,—each of them requiring every computation to be done over again. But the result was satisfactory, for it showed that the residuals were most nearly accounted for by a distance not far from 45 units (or a little less if the planet was at the opposite side of its orbit), and that the residuals increased for a distance greater or less than this. But still he was not satisfied, and for greater security he took up terms of the second and third order—very difficult to deal with—but found that they made no substantial difference in the result.

So much for the longitude of X (that is its orbit and position in the plane of the ecliptic) but that was not all, for its orbit might not lie in that plane but might be inclined to it, and like all the other planets he supposed it more or less so—more he surmised. Although he made some calculations on the subject he did not feel that any result obtained would be reliable, and if the longitude were near enough he thought the planet could be found. He says:

“To determine the inclination of the orbit of the unknown from the residuals in latitude of _Uranus_ has proved as inconclusive as Leverrier found the like attempt in the case of _Neptune_.

“The cause of failure lies, it would seem, in the fact that the elements of X enter into the observational equations for the latitude. Not only e and ῶ are thus initially affected but ε as well. Hence as these are doubtful from the longitude results, we can get from the latitude ones only doubtfulness to the second power.” Nevertheless he makes some calculations on the subject which, however, prove unsatisfactory.

Such in outline was his method of calculating the probable orbit and position in the sky of the trans-Neptunian planet; an herculean labor carried out with infinite pains, and attaining, not absolute definiteness, but results from the varying solutions sufficiently alike to warrant the belief in a close approximation. In dealing with what he calls the credentials for the acceptance of his results, he points out that one of his solutions for X in which he has much confidence, reduces the squares of the residuals to be accounted for by ninety per cent., and in the case of some of the others almost to nothing. Yet he had no illusions about the uncertainty of the result, for in the conclusions of the Memoir he says:

“But that the investigation opens our eyes to the pitfalls of the past does not on that account render us blind to those of the present. To begin with, the curves of the solutions show that a proper change in the errors of observation would quite alter the minimum point for either the different mean distances or the mean longitudes. A slight increase of the actual errors over the most probable ones, such as it by no means strains human capacity for error to suppose, would suffice entirely to change the most probable distance of the disturber and its longitude at the epoch. Indeed the imposing ‘probable error’ of a set of observations imposes on no one familiar with observation, the actual errors committed, due to systematic causes, always far exceeding it.

“In the next place the solutions themselves tell us of alternatives between which they leave us in doubt to decide. If we go by residuals alone, we should choose those solutions which have their mean longitudes at the epoch in the neighborhood of 0°, since the residuals are there the smallest. But on the other hand this would place the unknown now and for many decades back in a part of the sky which has been most assiduously scanned, while the solutions with ε around 180° lead us to one nearly inaccessible to most observatories, and, therefore, preferable for planetary hiding. Between the elements of the two, there is not much to choose, all agreeing pretty well with one another.

“Owing to the inexactitude of our data, then, we cannot regard our results with the complacency of completeness we should like.”

The bulk of the computations for the trans-Neptunian planet were finished by the spring of 1914, and in April he sent to Flagstaff from Boston, where the work had been done, two of the assistant computers. The final Memoir he read to the American Academy of Arts and Sciences on January 13, 1915; and printed in the spring as a publication of the Observatory. Naturally he was deeply anxious to see the fruit from such colossal labor. In July, 1913, he had written to Mr. Lampland: “Generally speaking what fields have you taken? Is there nothing suspicious?” and in May, 1914, “Don’t hesitate to startle me with a telegram ‘FOUND.’” Again, in August, he writes to Dr. Slipher: “I feel sadly of course that nothing has been reported about X, but I suppose the bad weather and Mrs. Lampland’s condition may somewhat explain it”; and to Mr. Lampland in December: “I am giving my work before the Academy on January 13. It would be thoughtful of you to announce the actual discovery at the same time.” Through the banter one can see the craving to find the long-sought planet, and the grief at the baffling of his hopes. That X was not found was the sharpest disappointment of his life.

If so much labor without tangible result gave little satisfaction, there was still less glory won by a vast calculation that did not prove itself correct. Curiously enough, he always enjoyed more recognition among astronomers in Europe than in America; for here, as a highly distinguished member of the craft recently remarked, he did not belong to the guild. He was fond of calling himself an amateur—by which he meant one who worked without remuneration—and of noting how many of the great contributors to science were in that category. The guild here was not readily hospitable to those who had not been trained in the regular treadmill; and it had been shocked by his audacity in proclaiming a discovery of intelligent handiwork on Mars. So for the most part he remained to the end of his life an amateur in this country; though what would have been said had he succeeded in producing, by rigorous calculation, an unknown planet far beyond the orbit of Neptune, it is interesting to conjecture, but difficult to know, for the younger generation of astronomers had not then come upon the stage nor the older ones outlived their prejudice.

The last eighteen months of his life were spent as usual partly at Flagstaff, where he was adding to the buildings, partly in Boston, and in lecturing. In May, 1916, he writes to Sig. Rigano of “Scientia” that he has not time to write an article for his Review, and adds: “Eventually I hope to publish a work on each planet—the whole connected together—but the end not yet.” Fortunately he did not know how near it was.

In May he lectured at Toronto; and in the autumn in the Northwest on Mars and other planets, at Washington State and Reed Colleges, and the universities of Idaho, Washington, Oregon and California. These set forth his latest views, often including much that had been discovered at Flagstaff and elsewhere since his earlier books were published; for his mind was far from closed to change of opinion on newly discovered evidence. It was something of a triumphal procession at these institutions; but it was too much.

More exhausted than he was himself aware, he returned to Flagstaff eager about a new investigation he had been planning on Jupiter’s satellites. It will be recalled that he had found the exact position of the gap in Saturn’s rings accounted for if the inner layers of the planet rotated faster and therefore were more oblate than the visible gaseous surface. Now the innermost satellite of Jupiter (the Vth) was farther off than the simple relation between distance and period should make it, a difference that might be explained if in Jupiter, as in Saturn, the molten inner core were more oblate than the outer gaseous envelope. To ascertain this the distance of the satellite V. must be determined exactly, and with Mr. E. C. Slipher he was busy in doing so night after night through that of November 11th. But he was overstrained, and the next day, November 12, 1916, not long after his return to Flagstaff, an attack of apoplexy brought to a sudden close his intensely active life. Before he became unconscious he said that he always knew it would come thus, but not so soon.

He lies buried in a mausoleum built by his widow close to the dome where his work was done.

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