esult,--the solution of what he calls "the great problem of
exact astronomy," the theoretical explanation of the observed motions of
the heavenly bodies.

If the universe consisted of but two bodies,--say, the sun and a
planet,--the motion would be simplicity itself; the planet would describe
an exact ellipse about the sun, and this orbit would never change in form,
size, or position. With the addition of only one more body, the problem at
once becomes so much more difficult as to be practically insoluble;
indeed, the "problem of the three bodies" has been attacked by astronomers
for years without the discovery of any general formula to express the
resulting motions. For the actually existing system of many planets with
their satellites and countless asteroids, only an approximation is
possible. The actual motions as observed and measured from year to year
are most complex. Can these be completely accounted for by the mutual
attractions of the bodies, according to the law of gravitation as
enunciated by Sir Isaac Newton? In Newcomb's words, "Does any world move
otherwise than as it is attracted by other worlds?" Of course, Newcomb has
not been the only astronomer at work on this problem, but it has been his
life-work and his contributions to its solution have been very noteworthy.

It is difficult to make the ordinary reader understand the obstacles in
the way of such a determination as this. Its two elements are, of course,
the mapping out of the lines in which the bodies concerned actually do
move and the calculations of the orbits in which they ought to move, if
the accepted laws of planetary motion are true. The first involves the
study of thousands of observations made during long years by different men
in far distant lands, the discussion of their probable errors, and their
reduction to a common standard. The latter requires the use of the most
refined methods of mathematical analysis; it is, as Newcomb says, "of a
complexity beyond the powers of ordinary conception." In works on
celestial mechanics a single formula may fill a whole chapter.

This problem first attracted Newcomb's attention when a young man at
Cambridge, when by analysis of the motions of the asteroids he showed that
the orbits of these minor planets had not, for several hundred thousand
years past, intersected at a single point, and that they could not,
therefore, have resulted, during that period, from the explosion of a
single large body, as h