In passing, I may remark that the failure of Galileo to ascertain the real shape of these appendages has always seemed to me to afford striking evidence of the importance of careful reasoning upon all observations whose actual significance is not at once apparent. If Galileo had been thus careful to analyse his observations of Saturn, he could not have failed to ascertain their real meaning. He had seen the planet apparently attended by two large satellites, one on either side, 'as though supporting the aged Saturn upon his slow course around the sun.' Night after night he had seen these attendants, always similarly placed, one on either side of the planet, and at equal distances from it. Then in 1612 he had again examined the planet, and lo, the attendants had vanished, 'as though Saturn had been at his old tricks, and had devoured his children.' But after a while the attendant orbs had reappeared in their former positions, had seemed slowly to grow larger, until at length they had presented the appearance of two pairs of mighty arms encompassing the planet. If Galileo had reasoned upon these changes of appearance, he could not have failed, as it seems to me, to interpret their true meaning. The three forms under which the rings had been seen by him sufficed to indicate the true shape of the appendage. Because Saturn was seen with two attendants of apparently equal size and always equi-distant from him, it was certain that there must be some appendage surrounding him, and extending to that distance from his globe. Because this appendage disappeared, it was certain that it must be thin and flat. Because it appeared at another time with a dark space between the arms and the planet, it was certain that the appendage is separated by a wide gap from the body of the planet. So that Galileo might have concluded--not doubtfully, but with assured confidence--that the appendage is a thin flat ring nowhere attached to the planet, or, as Huyghens said some forty years later, Saturn '_annulo cingitur tenui, plano, nusquam cohærente_.' Whether such reasoning would have been accepted by the contemporaries of Galileo may be doubtful. The generality of men are not content with reasoning which is logically sound, but require evidence which they can easily understand. Very likely Huyghens' proof from direct observation, though in reality not a whit more complete and far rougher, would have been regarded as the first true proof of the existence of Saturn's ring, just as Sir W. Herschel's observation of one star actually moving round another was regarded as the first true proof of the physical association of certain stars, a fact which Michell had proved as completely and far more neatly half a century earlier, by a method, however, which was 'caviare to the general.'