The explanation they gave was the explanation still given by astronomers. Let us briefly consider it. In doing so I propose to modify the ordinary text-book illustration which has always seemed to me ingeniously calculated (with its double set of diversely illuminated moons around the earth) to make a simple subject obscure. In fig. 12, let E represent the earth one half in darkness, the other half illuminated by the rays of the sun S, which should be supposed placed at a much greater distance to the left,--in fact, about five yards away from E. To preserve the right proportions, also, the sun ought to be much smaller and the earth a mere point. I mention this to prevent the reader from adopting erroneous ideas as to the size of these bodies. In reality it is quite impossible to show in such figures the true proportions of the heavenly bodies and of their distances. Next let M_{1}, M_{2}, M_{3}, etc., represent the moon in different positions along her circuit around the earth at E. [Illustration: Fig. 12.--Explaining the Moon's changes.] [Illustration: Fig. 13.--Illustrating the Moon's changes.] Now, it is clear that when the moon is at M_{1}, her illuminated face is turned from the earth, E. She therefore cannot be seen; and accordingly, in fig. 12, she is presented as a black disc at 1 to correspond with her invisibility when she is as at M_{1}. She passes on to M_{2}; and now from E a part of her illuminated half can be seen towards the sun, which would be towards the right, if we imagine an eye at E looking towards M_{2}. Her appearance then is as shown at 2, fig. 13. In any intermediate portion between M_{1} and M_{2}, the sickle of light is visible but narrower. We see also that all this time the moon's place on the sky cannot be far from the sun's place, for the line from E to M_{2} is not greatly inclined to the line from E to S. When the moon has got round to M_{3}, the observer on the earth sees as much of the dark half as of the bright half of the moon, the bright half being seen, of course, towards the sun. Thus the moon appears as at 3, fig. 13, Again as to position, the moon is now a quarter of a circuit of the heavens from the sun, for the line from E to M_{3} is square to the line from E to S. We see similarly that when at M_{4} the moon appears as shown at 4, fig. 13, for now the observer at E sees as small a part of the moon's dark side as he had seen of her bright side when she was at M_{2}. When she is at M_{5} the observer at E sees her bright face only, the dark face being turned directly from him. She, therefore, appears as at 5, fig. 13. Also being now exactly opposite the sun, as we see from fig. 12, she is at her highest when the sun is at his lowest, or at midnight; and at this time she rules the night as the sun rules the day.[10] As the moon passes on to M_{6}, a portion of her dark half comes into view, the bright side being now towards the left, as we look at M_{5} from E, fig. 12.