Correct 1820 1835 1872 1830 1861 1856

Within +-5 deg. {1812 1827 1865 1813 1815 1826
{1827 1842 1877 1866 1871 1868

Now the date at which it was most important to obtain the correct place
was 1845 or thereabouts when it was proposed to look for the planet; but
no special precaution seems to have been taken by either investigator to
secure any advantage for this particular date. Criticising the procedure
after the event (and of course this is a very unsatisfactory method of
criticism), we should say that it would have been better to make several
assumptions as regards the distance instead of relying upon Bode's Law;
but no one, so far as I know, has ever taken the trouble to write out a
satisfactory solution of the problem as it might have been conducted. Such
a solution would be full of interest, though it could only have a small
weight in forming our estimation of the skill with which the problem was
solved in the first instance.

[Sidenote: Le Verrier's erroneous limits.]

Fourthly, we may notice a very curious point. Le Verrier went to some
trouble not only to point out the most likely place for the planet, but to
indicate limits outside which it was not necessary to look. This part of
his work is specially commented upon with enthusiasm by Airy, and I will
reproduce what he says. It is rather technical perhaps, but those who
cannot follow the mathematics will be able to appreciate the tone of
admiration.

[Sidenote: The visible disc.]

"M. Le Verrier then enters into a most ingenious computation of the
limits between which the planet must be sought. The principle is
this: assuming a time of revolution, all the other unknown
quantities may be varied in such a manner that though the
observations will not be so well represented as before, yet the
errors of observation will be tolerable. At last, on continuing the
variation of elements, one error of observation will be intolerably
great. Then, by varying the elements in another way, we may at length
make another error of observation intolerably great; and so on. If we
compute, for all these different varieties of elements, the place of
the planet for 1847, its _locus_ will evidently be a discontinuous
curve or curvilinear polygon. If we do the same thing with different
periodic times, we shall get different polygons; and the extreme
per