"Neptune cannot, then, be the planet of M. Le Verrier's theory, and cannot account for the observed perturbations of Uranus under the form of the inequalities involved in his analysis"--(_Proc. Amer. Acad. I._, 1846-1848, _p._ 66). [Sidenote: Newcomb's criticism.] At the time when Professor Peirce wrote, the orbit of Neptune was not sufficiently well determined to decide whether one of the two limitations might not be correct, though he could see that they could not both be right, and we now know that they are _both wrong_. The mean distance of Neptune is 30, which does _not_ lie between 35 and 37.9; and the longitude in 1800 was 225°, which does _not_ lie between 243° and 252°. The ingenious 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