found by which the positions of these orbits could
be determined at any time in the past, even hundreds of thousands of
years back. The general methods of doing this were known, but no
one had applied them to the especial case of these little planets.
Here, then, was an opportunity of tracing back the changes in these
orbits through thousands of centuries in order to find whether, at
a certain epoch in the past, so great a cataclysm had occurred as
the explosion of a world. Were such the case, it would be possible
almost to set the day of the occurrence. How great a feat would it
be to bring such an event at such a time to light!

I soon found that the problem, in the form in which it had been
attacked by previous mathematicians, involved no serious difficulty.
At the Springfield meeting of the American Association for the
Advancement of Science, in 1859, I read a paper explaining the
method, and showed by a curve on the blackboard the changes
in the orbit of one of the asteroids for a period, I think, of
several hundred thousand years,--"beyond the memory of the oldest
inhabitants"--said one of the local newspapers. A month later it was
extended to three other asteroids, and the result published in the
"Astronomical Journal." In the following spring, 1860, the final
results of the completed work were communicated to the American
Academy of Arts and Sciences in a paper "On the Secular Variations
and Mutual Relations of the Orbits of the Asteroids." The question
of the possible variations in the orbits and the various relations
amongst them were here fully discussed. One conclusion was that,
so far as our present theory could show, the orbits had never passed
through any common point of intersection.

The whole trend of thought and research since that time has been
toward the conclusion that no such cataclysm as that looked for ever
occurred, and that the group of small planets has been composed of
separate bodies since the solar system came into existence. It was,
of course, a great disappointment not to discover the cataclysm, but
next best to finding a thing is showing that it is not there. This,
it may be remarked, was the first of my papers to attract especial
notice in foreign scientific journals, though I had already published
several short notes on various subjects in the "Astronomical Journal."

At this point I may say something of the problems of mathematical
astronomy in the middle of the last cen