process which Airy admired and which Peirce himself calls
"consummately skilful" was wrong in principle. As Professor Newcomb has
said, "the error was the elementary one that, instead of considering all
the elements simultaneously variable, Le Verrier took them one at a time,
considering the others as fixed, and determining the limits between which
each could be contained on this hypothesis. No solver of least square
equations at the present day ought to make such a blunder. Of course one
trouble in Le Verrier's demonstration, had he attempted a rigorous one,
would have been the impossibility of forming the simultaneous equations
expressive of possible variations of all the elements."

[Sidenote: Element of good fortune.]

[Sidenote: The map used by Galle.]

The account of Le Verrier's limits by Professor Peirce, though it exhibits
the error with special clearness, is a little unfair to Le Verrier in one
point. If, instead of taking the limits for the date 1800, we take them
for 1846 (when the search for Neptune was actually made), we shall find
that they do include the actual place of the planet, as Airy found. The
erroneous mean motion of Le Verrier's planet allowed of his being right at
one time and wrong at another; and Airy examined the limits under
favourable conditions, which explains his enthusiasm. But we can scarcely
wonder that Professor Peirce came to the conclusion that the planet
discovered was not the one pointed out by Le Verrier, and had been found
by mere accident. And all these circumstances inevitably contribute to a
general impression that the calculators had a large element of good
fortune to thank for their success. Nor need we hesitate to make this
admission, for there is an element of good fortune in all discoveries. To
look no further than this--if a man had not been doing a particular thing
at a particular time, as he might easily not have been, most discoveries
would never have been made. If Sir William Herschel had not been looking
at certain small stars for a totally different purpose he would never have
found Uranus; and no one need hesitate to admit the element of chance in
the finding of Neptune. It is well illustrated by a glance at the map
which, as has been remarked, Galle used to compare with the sky on the
night when he made the actual discovery. The planet was found down near
the bottom corner of the map, and since the limits assigned for its place
might easily have varied a few d