There are on the heavens many hundreds of thousands of stars, and the problem of identifying the planet, if indeed it should lie among these stars, seemed a very complex matter.  Of course it is the abundant presence of the stars which causes the difficulty.  If the stars could have been got rid of, a sweep over the heavens would at once disclose all the planets which are bright enough to be visible with the telescopic power employed.  It is the fortuitous resemblance of the planet to the stars which enables it to escape detection.  To discriminate the planet among stars everywhere in the sky would be almost impossible.  If, however, some method could be devised for localizing that precise region in which the planet’s existence might be presumed, then the search could be undertaken with some prospect of success.

To a certain extent the problem of localizing the region on the sky in which the planet might be expected admitted of an immediate limitation.  It is known that all the planets, or perhaps I ought rather to say, all the great planets, confine their movements to a certain zone around the heavens.  This zone extends some way on either side of that line called the ecliptic in which the earth pursues its journey around the sun.  It was therefore to be inferred that the new planet need not be sought for outside this zone.  It is obvious that this consideration at once reduces the area to be scrutinized to a small fraction of the entire heavens.  But even within the zone thus defined there are many thousands of stars.  It would seem a hopeless task to detect the new planet unless some further limitation to its position could be assigned.

It was accordingly suggested to Le Verrier that he should endeavour to discover in what particular part of the strip of the celestial sphere which we have indicated the search for the unknown planet should be instituted.  The materials available to the mathematician for the solution of this problem were to be derived solely from the discrepancies between the calculated places in which Uranus should be found, taking into account the known causes of disturbance, and the actual places in which observation had shown the planet to exist.  Here was indeed an unprecedented problem, and one of extraordinary difficulty.  Le Verrier, however, faced it, and, to the astonishment of the world, succeeded in carrying it through to a brilliant solution.  We cannot here attempt to enter into any account of the mathematical investigations that were necessary.  All that we can do is to give a general indication of the method which had to be adopted.