on of mathematical research is shown by the
fact that we can now add up, as it were, all these momentary effects
through years and centuries, with a view of determining the combined
result at any one moment. It is true that this can be done only in an
imperfect way, and at the expense of enormous labor; but, by putting
more and more work into it, investigating deeper and deeper, taking
into account smaller and smaller terms of our formulae, and searching
for the minutest effects, we may gradually approach, though we may
never reach, absolute exactness. Here we see the first difficulty
in reaching a definite conclusion. One cannot be quite sure that
a deviation is not due to some imperfection in mathematical method
until he and his fellows have exhausted the subject so thoroughly
as to show that no error is possible. This is hard indeed to do.

Taking up the question on the observational side, a source of
difficulty and confusion at once presented itself. The motions
of a heavenly body from day to day and year to year are mapped out
by comparative observations on it and on the stars. The question
of the exact positions of the stars thus comes in. In determining
these positions with the highest degree of precision, a great variety
of data have to be used. The astronomer cannot reach a result by a
single step, nor by a hundred steps. He is like a sculptor chiseling
all the time, trying to get nearer and nearer the ideal form of his
statue, and finding that with every new feature he chisels out,
a defect is brought to light in other features. The astronomer,
when he aims at the highest mathematical precision in his results,
finds Nature warring with him at every step, just as if she wanted
to make his task as difficult as possible. She alters his personal
equation when he gets tired, makes him see a small star differently
from a bright one, gives his instrument minute twists with heat and
cold, sends currents of warm or cold air over his locality, which
refract the rays of light, asks him to keep the temperature in which
he works the same as that outside, in order to avoid refraction when
the air enters his observing room, and still will not let him do it,
because the walls and everything inside the room, being warmed up
during the day, make the air warmer than it is outside. With all
these obstacles which she throws in his way he must simply fight the
best he can, exerting untiring industry to eliminate their e