Once this tremendous step had been taken, the great difficulties which beset the monstrous conception of the celestial sphere vanished, for the stars need no longer be regarded as situated at equal distances from the earth.  Copernicus saw that they might lie at the most varied degrees of remoteness, some being hundreds or thousands of times farther away than others.  The complicated structure of the celestial sphere as a material object disappeared altogether; it remained only as a geometrical conception, whereon we find it convenient to indicate the places of the stars.  Once the Copernican doctrine had been fully set forth, it was impossible for anyone, who had both the inclination and the capacity to understand it, to withhold acceptance of its truth.  The doctrine of a stationary earth had gone for ever.

Copernicus having established a theory of the celestial movements which deliberately set aside the stability of the earth, it seemed natural that he should inquire whether the doctrine of a moving earth might not remove the difficulties presented in other celestial phenomena.  It had been universally admitted that the earth lay unsupported in space.  Copernicus had further shown that it possessed a movement of rotation.  Its want of stability being thus recognised, it seemed reasonable to suppose that the earth might also have some other kinds of movements as well.  In this, Copernicus essayed to solve a problem far more difficult than that which had hitherto occupied his attention.  It was a comparatively easy task to show how the diurnal rising and setting could be accounted for by the rotation of the earth.  It was a much more difficult undertaking to demonstrate that the planetary movements, which Ptolemy had represented with so much success, could be completely explained by the supposition that each of those planets revolved uniformly round the sun, and that the earth was also a planet, accomplishing a complete circuit of the sun once in the course of a year.

[Plate:  Explanation of planetary movements.]

It would be impossible in a sketch like the present to enter into any detail as to the geometrical propositions on which this beautiful investigation of Copernicus depended.  We can only mention a few of the leading principles.  It may be laid down in general that, if an observer is in movement, he will, if unconscious of the fact, attribute to the fixed objects around him a movement equal and opposite to that which he actually possesses.  A passenger on a canal-boat sees the objects on the banks apparently moving backward with a speed equal to that by which he is himself advancing forwards.  By an application of this principle, we can account for all the phenomena of the movements of the planets, which Ptolemy had so ingeniously represented by his circles.  Let us take, for instance, the most characteristic feature in the irregularities of the outer planets.  We have already remarked that