iodic times that can be allowed will be indicated by the polygons
becoming points. These extreme periodic times are 207 and 233 years.
If now we draw one grand curve, circumscribing all the polygons, it
is certain that the planet must be within that curve. In one
direction, M. Le Verrier found no difficulty in assigning a limit; in
the other he was obliged to restrict it, by assuming a limit to the
eccentricity. Thus he found that the longitude of the planet was
certainly not less than 321 deg., and not greater than 335 deg. or 345 deg.,
according as we limit the eccentricity to 0.125 or 0.2. And if we
adopt 0.125 as the limit, then the mass will be included between the
limits 0.00007 and 0.00021; either of which exceeds that of _Uranus_.
From this circumstance, combined with a probable hypothesis as to the
density, M. Le Verrier concluded that the planet would have a
visible disk, and sufficient light to make it conspicuous in ordinary
telescopes.

"M. Le Verrier then remarks, as one of the strong proofs of the
correctness of the general theory, that the error of radius vector is
explained as accurately as the error of longitude. And finally, he
gives his opinion that the latitude of the disturbing planet must be
small.

"My analysis of this paper has necessarily been exceedingly
imperfect, as regards the astronomical and mathematical parts of it;
but I am sensible that, in regard to another part, it fails totally.
I cannot attempt to convey to you the impression which was made on me
by the author's undoubting confidence in the general truth of his
theory, by the calmness and clearness with which he limited the field
of observation, and by the firmness with which he proclaimed to
observing astronomers, 'Look in the place which I have indicated, and
you will see the planet well.' Since Copernicus declared that, when
means should be discovered for improving the vision, it would be
found that _Venus_ had phases like the moon, nothing (in my opinion)
so bold, and so justifiably bold, has been uttered in astronomical
prediction. It is here, if I mistake not, that we see a character far
superior to that of the able, or enterprising, or industrious
mathematician; it is here that we see the philosopher."

[Sidenote: Peirce's views of the limits.]

But now t