Examine Fig. 67. Here A B G D represent the ecliptic, E its centre, and A E G its diameter; Z H T K is the epicycle, in which the planet moves uniformly around the centre A, while the epicycle uniformly traverses the circle A B G D. Now, suppose that the star has arrived at H; it would appear to an observer at E to be more advanced by the uniform movement of all the arc A H; if it be at K, it would appear, on the contrary, to be less advanced by all the arc A K. At Z the star would appear more distant, and at T, nearer than if it were at A. [Illustration: FIG. 67.--The Circle and the Epicycle.] * * * * * To explain the other phenomena, such as the stations and retrocessions of the planets, recourse was again had to the epicycles or deferred eccentric circles. By multiplying these it was possible to account for all the angular inequalities in the movements of a planet. It is of importance to note this point, in order to show how very dangerous it is to trust absolutely to mathematics in our search after the truth; that science which, by the certainty of its demonstrations, nourishes our intellectual pride, and may, therefore, occasionally lull the mind into a false security. The theory of epicycles, from a mathematical point of view, was irreproachable, and it sufficiently accounted for the facts which threatened to overthrow the dogma of circular orbits and uniform planetary movement. * * * * * But by degrees, as observations grew more accurate and comprehensive, these and other theories, however fine in appearance,--_teres atque rotundus_,--gradually disappeared, if fundamentally erroneous. By the invention of micrometers, we were enabled to measure more exactly than had formerly been possible the variations of diameter or the modifications of distance, and afterwards to compare them with the changes of velocity. From this comparison it results that the latter are not greater than is compatible with the alterations of distance indicated by the variations of diameter; in a word, that the hypothesis of epicycles is decidedly insufficient to account for all the inequalities detected by careful investigation. * * * * * Kepler was the first to break the charm which had held captive the mind of astronomers, including even Copernicus and Tycho Brahé. Ptolemæus had considered the mean positions of the stars to be real. Kepler, strong in his researches, declared that they were but a factitious mode of calculation by which the _true_ positions might be ascertained; that the mean movement is simply an artifice representing the star's place, _if_ no inequality existed; in fine, that we must take the movements as they are in nature,--the true movements, given by observation,--and not the mean movements, deduced from an erroneous hypothesis.