zzle by the theory of gravitation was hailed with the
greatest enthusiasm by astronomers, and it established the fame of
the two French mathematicians.

Next they attacked the complicated problem of the motions of Jupiter's
satellites. They succeeded in obtaining a theory of their motions which
represented fact very nearly indeed, and they detected the following
curious relationship between the satellites:--The speed of the first
satellite + twice the speed of the second is equal to the speed of the
third.

They found this, not empirically, after the manner of Kepler, but as a
deduction from the law of gravitation; for they go on to show that even
if the satellites had not started with this relation they would sooner
or later, by mutual perturbation, get themselves into it. One singular
consequence of this, and of another quite similar connection between
their positions, is that all three satellites can never be eclipsed at
once.

The motion of the fourth satellite is less tractable; it does not so
readily form an easy system with the others.

After these great successes the two astronomers naturally proceeded to
study the mutual perturbations of all other bodies in the solar system.
And one very remarkable discovery they made concerning the earth and
moon, an account of which will be interesting, though the details and
processes of calculation are quite beyond us in a course like this.

Astronomical theory had become so nearly perfect by this time, and
observations so accurate, that it was possible to calculate many
astronomical events forwards or backwards, over even a thousand years or
more, with admirable precision.

Now, Halley had studied some records of ancient eclipses, and had
calculated back by means of the lunar theory to see whether the
calculation of the time they ought to occur would agree with the record
of the time they did occur. To his surprise he found a discrepancy, not
a large one, but still one quite noticeable. To state it as we know it
now:--An eclipse a century ago happened twelve seconds later than it
ought to have happened by theory; two centuries back the error amounted
to forty-eight seconds, in three centuries it would be 108 seconds, and
so on; the lag depending on the square of the time. By research, and
help from scholars, he succeeded in obtaining the records of some very
ancient eclipses indeed. One in Egypt towards the end of the tenth
century A.D.; another in 201 A.D