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 analytic
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 analysis 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 quantities 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 circumstances 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 class 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 quantities 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 hitherto 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 Uranus 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
latitude 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 hitherto 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 mass 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 class 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 masses 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
compass. 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 compass 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 compass 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 passage, so that the
captain could not get a sight on the sun, and therefore had to trust to
his compass 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 ashore 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 latitude 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 latitude
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 latitude and longitude within fifty yards, the
positions of the principal 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
orbit. 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 principal 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 space 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 gods 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 orbit was
exactly between two stars, when it moved to the other side of its orbit
it would not be in a line between them, but each star would have seemed
to move in the opposite direction.