But by his theory the planets must attract one another and these attractions must cause the motions of each to deviate slightly from the laws in question. Since such deviations were actually observed it was very natural to conclude that they were due to this cause, but how shall we prove it? To do this with all the rigor required in a mathematical investigation it is necessary to calculate the effect of the mutual action of the planets in changing their orbits. This calculation must be made with such precision that there shall be no doubt respecting the results of the theory. Then its results must be compared with the best observations. If the slightest outstanding difference is established there is something wrong and the requirements of astronomical science are not satisfied. The complete solution of this problem was entirely beyond the power of Newton. When his methods of research were used he was indeed able to show that the mutual action of the planets would produce deviations in their motions of the same general nature with those observed, but he was not able to calculate these deviations with numerical exactness. His most successful attempt in this direction was perhaps made in the case of the moon. He showed that the sun"s disturbing force on this body would produce several inequalities the existence of which had been established by observation, and he was also able to give a rough estimate of their amount, but this was as far as his method could go. A great improvement had to be made, and this was effected not by English, but by continental mathematicians.
The latter saw, clearly, that it was impossible to effect the required solution by the geometrical mode of reasoning employed by Newton. The problem, as it presented itself to their minds, was to find algebraic expressions for the positions of the planets at any time. The lat.i.tude, longitude, and radius-vector of each planet are constantly varying, but they each have a determined value at each moment of time. They may therefore be regarded as functions of the time, and the problem was to express these functions by algebraic formulae. These algebraic expressions would contain, besides the time, the elements of the planetary orbits to be derived from observation. The time which we may suppose to be represented algebraically by the symbol t, would remain as an unknown quant.i.ty to the end. What the mathematician sought to do was to present the astronomer with a series of algebraic expressions containing t as an indeterminate quant.i.ty, and so, by simply subst.i.tuting for t any year and fraction of a year whatever--1600, 1700, 1800, for example, the result would give the lat.i.tude, longitude, or radius-vector of a planet.
The problem as thus presented was one of the most difficult we can perceive of, but the difficulty was only an incentive to attacking it with all the greater energy. So long as the motion was supposed purely elliptical, so long as the action of the planets was neglected, the problem was a simple one, requiring for its solution only the a.n.a.lytic geometry of the ellipse. The real difficulties commenced when the mutual action of the planets was taken into account. It is, of course, out of the question to give any technical description or a.n.a.lysis of the processes which have been invented for solving the problem; but a brief historical sketch may not be out of place. A complete and rigorous solution of the problem is out of the question--that is, it is impossible by any known method to form an algebraic expression for the co-ordinates of a planet which shall be absolutely exact in a mathematical sense. In whatever way we go to work the expression comes out in the form of an infinite series of terms, each term being, on the whole, a little smaller as we increase the number. So, by increasing the number of these various terms, we can approach nearer and nearer to a mathematical exactness, but can never reach it. The mathematician and astronomer have to be satisfied when they have carried the solution so far that the neglected quant.i.ties are entirely beyond the powers of observation.
Mathematicians have worked upon the problem in its various phases for nearly two centuries, and many improvements in detail have, from time to time, been made, but no general method, applicable to all cases, has been devised. One plan is to be used in treating the motion of the moon, another for the interior planets, another for Jupiter and Saturn, another for the minor planets, and so on. Under these circ.u.mstances it will not surprise you to learn that our tables of the celestial motions do not, in general, correspond in accuracy to the present state of practical astronomy. There is no authority and no office in the world whose duty it is to look after the preparations of the formulae I have described. The work of computing them has been almost entirely left to individual mathematicians whose taste lay in that direction, and who have sometimes devoted the greater part of their lives to calculations on a single part of the work. As a striking instance of this, the last great work on the Motion of the Moon, that of Delaunay, of Paris, involved some fifteen years of continuous hard labor.
Hansen, of Germany, who died five years ago, devoted almost his whole life to investigations of this cla.s.s and to the development of new methods of computation. His tables of the moon are those now used for predicting the places of the moon in all the ephemerides of the world.
The only successful attempt to prepare systematic tables for all the large planets is that completed by Le Verrier just before his death; but he used only a small fraction of the material at his disposal, and did not employ the modern methods, confining himself wholly to those invented by his countrymen about the beginning of the present century.
For him Jacobi and Hansen had lived in vain.
The great difficulty which besets the subject arises from the fact that mathematical processes alone will not give us the position of a planet, there being seven unknown quant.i.ties for each planet which must be determined by observations. A planet, for instance, may move in any ellipse whatever, having the sun in one focus, and it is impossible to tell what ellipse it is, except from observation. The mean motion of a planet, or its period of revolution, can only be determined by a long series of observations, greater accuracy being obtained the longer the observations are continued. Before the time of Bradley, who commenced work at the Greenwich Observatory about 1750, the observations were so far from accurate that they are now of no use whatever, unless in exceptional cases. Even Bradley"s observations are in many cases far less accurate than those made now. In consequence, we have not heretofore had a sufficiently extended series of observations to form an entirely satisfactory theory of the celestial motions.
As a consequence of the several difficulties and drawbacks, when the computation of our ephemeris was started, in the year 1849, there were no tables which could be regarded as really satisfactory in use. In the British Nautical Almanac the places of the moon were derived from the tables of Burckhardt published in the year 1812. You will understand, in a case like this, no observations subsequent to the issue of the tables are made use of; the place of the moon of any day, hour, and minute of Greenwich time, mean time, was precisely what Burckhardt would have computed nearly a half a century before. Of the tables of the larger planets the latest were those of Bouvard, published in 1812, while the places of Venus were from tables published by Lindenau in 1810. Of course such tables did not possess astronomical accuracy. At that time, in the case of the moon, completely new tables were constructed from the results reached by Professor Airy in his reduction of the Greenwich observations of the moon from 1750 to 1830. These were constructed under the direction of Professor Pierce and represented the places of the moon with far greater accuracy than the older tables of Burckhardt. For the larger planets corrections were applied to the older tables to make them more nearly represent observations before new ones were constructed. These corrections, however, have not proved satisfactory, not being founded on sufficiently thorough investigations. Indeed, the operation of correcting tables by observation, as we would correct the dead-reckoning of a ship, is a makeshift, the result of which must always be somewhat uncertain, and it tends to destroy that unity which is an essential element of the astronomical ephemeris designed for permanent future use. The result of introducing them, while no doubt an improvement on the old tables, has not been all that should be desired. The general lack of unity in the tables. .h.i.therto employed is such that I can only state what has been done by mentioning each planet in detail.
For Mercury, new tables were constructed by Professor Winlock, from formulae published by Le Verrier in 1846. These tables have, however, been deviating from the true motion of the planet, owing to the motion of the perihelion of Mercury, subsequently discovered by Le Verrier himself. They are now much less accurate than the newer tables published by Le Verrier ten years later.
Of Venus new tables were constructed by Mr. Hill in 1872. They are more accurate than any others, being founded on later data than those of Le Verrier, and are therefore satisfactory so far as accuracy of prediction is concerned.
The place of Mars, Jupiter, and Saturn are still computed from the old tables, with certain necessary corrections to make them better represent observations.
The places of Ura.n.u.s and Neptune are derived from new tables which will probably be sufficiently accurate for some time to come.
For the moon, Pierce"s tables have been employed up to the year 1882 inclusive. Commencing with the ephemeris for the year 1883, Hansen"s tables are introduced with corrections to the mean longitude founded on two centuries of observation.
With so great a lack of uniformity, and in the absence of any existing tables which have any other element of unity than that of being the work of the same authors, it is extremely desirable that we should be able to compute astronomical ephemerides from a single uniform and consistent set of astronomical data. I hope, in the course of years, to render this possible.
When our ephemeris was first commenced, the corrections applied to existing tables rendered it more accurate than any other. Since that time, the introduction into foreign ephemerides of the improved tables of Le Verrier have rendered them, on the whole, rather more accurate than our own. In one direction, however, our ephemeris will hereafter be far ahead of all others. I mean in its positions of the fixed stars.
This portion of it is of particular importance to us, owing to the extent to which our government is engaged in the determination of positions on this continent, and especially in our western territories.
Although the places of the stars are determined far more easily than those of the planets, the discussion of star positions has been in almost as backward a state as planetary positions. The errors of old observers have crept in and been continued through two generations of astronomers. A systematic attempt has been made to correct the places of the stars for all systematic errors of this kind, and the work of preparing a catalogue of stars which shall be completely adapted to the determination of time and longitude, both in the fixed observatory and in the field, is now approaching completion. The catalogue cannot be sufficiently complete to give places of the stars for determining the lat.i.tude by the zenith telescope, because for such a purpose a much greater number of stars is necessary than can be incorporated in the ephemeris.
From what I have said, it will be seen that the astronomical tables, in general, do not satisfy the scientific condition of completely representing observations to the last degree of accuracy. Few, I think, have an idea how unsystematically work of this kind has. .h.i.therto been performed. Until very lately the tables we have possessed have been the work of one man here, another there, and another one somewhere else, each using different methods and different data. The result of this is that there is nothing uniform and systematic among them, and that they have every range of precision. This is no doubt due in part to the fact that the construction of such tables, founded on the ma.s.s of observation hitherto made, is entirely beyond the power of any one man.
What is wanted is a number of men of different degrees of capacity, all co-operating on a uniform system, so as to obtain a uniform result, like the astronomers in a large observatory. The Greenwich Observatory presents an example of co-operative work of this cla.s.s extending over more than a century. But it has never extended its operations far outside the field of observation, reduction, and comparison with existing tables. It shows clearly, from time to time, the errors of the tables used in the British Nautical Almanac, but does nothing further, occasional investigations excepted, in the way of supplying new tables.
An exception to this is a great work on the theory of the moon"s motion, in which Professor Airy is now engaged.
It will be understood that several distinct conditions not yet fulfilled are desirable in astronomical tables; one is that each set of tables shall be founded on absolutely consistent data, for instance, that the ma.s.ses of the planets shall be the same throughout. Another requirement is that this data shall be as near the truth as astronomical data will suffice to determine them. The third is that the results shall be correct in theory. That is, whether they agree or disagree with observations, they shall be such as result mathematically from the adopted data.
Tables completely fulfilling these conditions are still a work of the future. It is yet to be seen whether such co-operation as is necessary to their production can be secured under any arrangement whatever.
XIV
THE WORLD"S DEBT TO ASTRONOMY
Astronomy is more intimately connected than any other science with the history of mankind. While chemistry, physics, and we might say all sciences which pertain to things on the earth, are comparatively modern, we find that contemplative men engaged in the study of the celestial motions even before the commencement of authentic history.
The earliest navigators of whom we know must have been aware that the earth was round. This fact was certainly understood by the ancient Greeks and Egyptians, as well as it is at the present day. True, they did not know that the earth revolved on its axis, but thought that the heavens and all that in them is performed a daily revolution around our globe, which was, therefore, the centre of the universe. It was the cynosure, or constellation of the Little Bear, by which the sailors used to guide their ships before the discovery of the mariner"s compa.s.s. Thus we see both a practical and contemplative side to astronomy through all history. The world owes two debts to that science: one for its practical uses, and the other for the ideas it has afforded us of the immensity of creation.
The practical uses of astronomy are of two kinds: One relates to geography; the other to times, seasons, and chronology. Every navigator who sails long out of sight of land must be something of an astronomer.
His compa.s.s tells him where are east, west, north, and south, but it gives him no information as to where on the wide ocean he may be, or whither the currents may be carrying him. Even with the swiftest modern steamers it is not safe to trust to the compa.s.s in crossing the Atlantic. A number of years ago the steamer City of Washington set out on her usual voyage from Liverpool to New York. By rare bad luck the weather was stormy or cloudy during her whole pa.s.sage, so that the captain could not get a sight on the sun, and therefore had to trust to his compa.s.s and his log-line, the former telling him in what direction he had steamed, and the latter how fast he was going each hour. The result was that the ship ran ash.o.r.e on the coast of Nova Scotia, when the captain thought he was approaching Nantucket.
Not only the navigator but the surveyor in the western wilds must depend on astronomical observations to learn his exact position on the earth"s surface, or the lat.i.tude and longitude of the camp which he occupies. He is able to do this because the earth is round, and the direction of the plumb-line not exactly the same at any two places. Let us suppose that the earth stood still, so as not to revolve on its axis at all. Then we should always see the stars at rest and the star which was in the zenith of any place, say a farm-house in New York, at any time, would be there every night and every hour of the year. Now the zenith is simply the point from which the plumb-line seems to drop. Lie on the ground; hang a plummet above your head, sight on the line with one eye, and the direction of the sight will be the zenith of your place. Suppose the earth was still, and a certain star was at your zenith. Then if you went to another place a mile away, the direction of the plumb-line would be slightly different. The change would, indeed, be very small, so small that you could not detect it by sighting with the plumb-line. But astronomers and surveyors have vastly more accurate instruments than the plumb-line and the eye, instruments by which a deviation that the unaided eye could not detect can be seen and measured. Instead of the plumb-line they use a spirit-level or a basin of quicksilver. The surface of quicksilver is exactly level and so at right angles to the true direction of the plumb-line or the force of gravity. Its direction is therefore a little different at two different places on the surface, and the change can be measured by its effect on the apparent direction of a star seen by reflection from the surface.
It is true that a considerable distance on the earth"s surface will seem very small in its effect on the position of a star. Suppose there were two stars in the heavens, the one in the zenith of the place where you now stand, and the other in the zenith of a place a mile away. To the best eye unaided by a telescope those two stars would look like a single one. But let the two places be five miles apart, and the eye could see that there were two of them. A good telescope could distinguish between two stars corresponding to places not more than a hundred feet apart. The most exact measurements can determine distances ranging from thirty to sixty feet. If a skilful astronomical observer should mount a telescope on your premises, and determine his lat.i.tude by observations on two or three evenings, and then you should try to trick him by taking up the instrument and putting it at another point one hundred feet north or south, he would find out that something was wrong by a single night"s work.
Within the past three years a wobbling of the earth"s axis has been discovered, which takes place within a circle thirty feet in radius and sixty feet in diameter. Its effect was noticed in astronomical observations many years ago, but the change it produced was so small that men could not find out what the matter was. The exact nature and amount of the wobbling is a work of the exact astronomy of the present time.
We cannot measure across oceans from island to island. Until a recent time we have not even measured across the continent, from New York to San Francisco, in the most precise way. Without astronomy we should know nothing of the distance between New York and Liverpool, except by the time which it took steamers to run it, a measure which would be very uncertain indeed. But by the aid of astronomical observations and the Atlantic cables the distance is found within a few hundred yards.
Without astronomy we could scarcely make an accurate map of the United States, except at enormous labor and expense, and even then we could not be sure of its correctness. But the practical astronomer being able to determine his lat.i.tude and longitude within fifty yards, the positions of the princ.i.p.al points in all great cities of the country are known, and can be laid down on maps.
The world has always had to depend on astronomy for all its knowledge concerning times and seasons. The changes of the moon gave us the first month, and the year completes its round as the earth travels in its...o...b..t. The results of astronomical observation are for us condensed into almanacs, which are now in such universal use that we never think of their astronomical origin. But in ancient times people had no almanacs, and they learned the time of year, or the number of days in the year, by observing the time when Sirius or some other bright star rose or set with the sun, or disappeared from view in the sun"s rays.
At Alexandria, in Egypt, the length of the year was determined yet more exactly by observing when the sun rose exactly in the east and set exactly in the west, a date which fixed the equinox for them as for us.
More than seventeen hundred years ago, Ptolemy, the great author of The Almagest, had fixed the length of the year to within a very few minutes. He knew it was a little less than 365 1/2 days. The dates of events in ancient history depend very largely on the chronological cycles of astronomy. Eclipses of the sun and moon sometimes fixed the date of great events, and we learn the relation of ancient calendars to our own through the motions of the earth and moon, and can thus measure out the years for the events in ancient history on the same scale that we measure out our own.
At the present day, the work of the practical astronomer is made use of in our daily life throughout the whole country in yet another way. Our fore-fathers had to regulate their clocks by a sundial, or perhaps by a mark at the corner of the house, which showed where the shadow of the house fell at noon. Very rude indeed was this method; and it was uncertain for another reason. It is not always exactly twenty-four hours between two noons by the sun, Sometimes for two or three months the sun will make it noon earlier and earlier every day; and during several other months later and later every day. The result is that, if a clock is perfectly regulated, the sun will be sometimes a quarter of an hour behind it, and sometimes nearly the same amount before it. Any effort to keep the clock in accord with this changing sun was in vain, and so the time of day was always uncertain.
Now, however, at some of the princ.i.p.al observatories of the country astronomical observations are made on every clear night for the express purpose of regulating an astronomical clock with the greatest exactness. Every day at noon a signal is sent to various parts of the country by telegraph, so that all operators and railway men who hear that signal can set their clock at noon within two or three seconds.
People who live near railway stations can thus get their time from it, and so exact time is diffused into every household of the land which is at all near a railway station, without the trouble of watching the sun.
Thus increased exactness is given to the time on all our railroads, increased safety is obtained, and great loss of time saved to every one. If we estimated the money value of this saving alone we should no doubt find it to be greater than all that our study of astronomy costs.
It must therefore be conceded that, on the whole, astronomy is a science of more practical use than one would at first suppose. To the thoughtless man, the stars seem to have very little relation to his daily life; they might be forever hid from view without his being the worse for it. He wonders what object men can have in devoting themselves to the study of the motions or phenomena of the heavens. But the more he looks into the subject, and the wider the range which his studies include, the more he will be impressed with the great practical usefulness of the science of the heavens. And yet I think it would be a serious error to say that the world"s greatest debt to astronomy was owing to its usefulness in surveying, navigation, and chronology. The more enlightened a man is, the more he will feel that what makes his mind what it is, and gives him the ideas of himself and creation which he possesses, is more important than that which gains him wealth. I therefore hold that the world"s greatest debt to astronomy is that it has taught us what a great thing creation is, and what an insignificant part of the Creator"s work is this earth on which we dwell, and everything that is upon it. That s.p.a.ce is infinite, that wherever we go there is a farther still beyond it, must have been accepted as a fact by all men who have thought of the subject since men began to think at all. But it is very curious how hard even the astronomers found it to believe that creation is as large as we now know it to be. The Greeks had their G.o.ds on or not very far above Olympus, which was a sort of footstool to the heavens. Sometimes they tried to guess how far it probably was from the vault of heaven to the earth, and they had a myth as to the time it took Vulcan to fall. Ptolemy knew that the moon was about thirty diameters of the earth distant from us, and he knew that the sun was many times farther than the moon; he thought it about twenty times as far, but could not be sure. We know that it is nearly four hundred times as far.
When Copernicus propounded the theory that the earth moved around the sun, and not the sun around the earth, he was able to fix the relative distances of the several planets, and thus make a map of the solar system. But he knew nothing about the scale of this map. He knew, for example, that Venus was a little more than two-thirds the distance of the earth from the sun, and that Mars was about half as far again as the earth, Jupiter about five times, and Saturn about ten times; but he knew nothing about the distance of any one of them from the sun. He had his map all right, but he could not give any scale of miles or any other measurements upon it. The astronomers who first succeeded him found that the distance was very much greater than had formerly been supposed; that it was, in fact, for them immeasurably great, and that was all they could say about it.
The proofs which Copernicus gave that the earth revolved around the sun were so strong that none could well doubt them. And yet there was a difficulty in accepting the theory which seemed insuperable. If the earth really moved in so immense an orbit as it must, then the stars would seem to move in the opposite direction, just as, if you were in a train that is shunting off cars one after another, as the train moves back and forth you see its motion in the opposite motion of every object around you. If then the earth at one side of its...o...b..t was exactly between two stars, when it moved to the other side of its...o...b..t it would not be in a line between them, but each star would have seemed to move in the opposite direction.
For centuries astronomers made the most exact observations that they were able without having succeeded in detecting any such apparent motion among the stars. Here was a mystery which they could not solve.
Either the Copernican system was not true, after all, and the earth did not move in an orbit, or the stars were at such immense distances that the whole immeasurable orbit of the earth is a mere point in comparison. Philosophers could not believe that the Creator would waste room by allowing the inconceivable s.p.a.ces which appeared to lie between our system and the fixed stars to remain unused, and so thought there must be something wrong in the theory of the earth"s motion.
Not until the nineteenth century was well in progress did the most skilful observers of their time, Bessel and Struve, having at command the most refined instruments which science was then able to devise, discover the reality of the parallax of the stars, and show that the nearest of these bodies which they could find was more than 400,000 times as far as the 93,000,000 of miles which separate the earth from the sun. During the half-century and more which has elapsed since this discovery, astronomers have been busily engaged in fathoming the heavenly depths. The nearest star they have been able to find is about 280,000 times the sun"s distance. A dozen or a score more are within 1,000,000 times that distance. Beyond this all is unfathomable by any sounding-line yet known to man.
The results of these astronomical measures are stupendous beyond conception. No mere statement in numbers conveys any idea of it. Nearly all the brighter stars are known to be flying through s.p.a.ce at speeds which generally range between ten and forty or fifty miles per second, some slower and some swifter, even up to one or two hundred miles a second. Such a speed would carry us across the Atlantic while we were reading two or three of these sentences. These motions take place some in one direction and some in another. Some of the stars are coming almost straight towards us. Should they reach us, and pa.s.s through our solar system, the result would be destructive to our earth, and perhaps to our sun.
Are we in any danger? No, because, however madly they may come, whether ten, twenty, or one hundred miles per second, so many millions of years must elapse before they reach us that we need give ourselves no concern in the matter. Probably none of them are coming straight to us; their course deviates just a hair"s-breadth from our system, but that hair"s-breadth is so large a quant.i.ty that when the millions of years elapse their course will lie on one side or the other of our system and they will do no harm to our planet; just as a bullet fired at an insect a mile away would be nearly sure to miss it in one direction or the other.
Our instrument makers have constructed telescopes more and more powerful, and with these the whole number of stars visible is carried up into the millions, say perhaps to fifty or one hundred millions. For aught we know every one of those stars may have planets like our own circling round it, and these planets may be inhabited by beings equal to ourselves. To suppose that our globe is the only one thus inhabited is something so unlikely that no one could expect it. It would be very nice to know something about the people who may inhabit these bodies, but we must await our translation to another sphere before we can know anything on the subject. Meanwhile, we have gained what is of more value than gold or silver; we have learned that creation transcends all our conceptions, and our ideas of its Author are enlarged accordingly.
XV
AN ASTRONOMICAL FRIENDSHIP
There are few men with whom I would like so well to have a quiet talk as with Father h.e.l.l. I have known more important and more interesting men, but none whose acquaintance has afforded me a serener satisfaction, or imbued me with an ampler measure of a feeling that I am candid enough to call self-complacency. The ties that bind us are peculiar. When I call him my friend, I do not mean that we ever hobn.o.bbed together. But if we are in sympathy, what matters it that he was dead long before I was born, that he lived in one century and I in another? Such differences of generation count for little in the brotherhood of astronomy, the work of whose members so extends through all time that one might well forget that he belongs to one century or to another.