observer from a, where he sees the moon rising, to b, where he sees it setting. When he is at c, on the line joining the centres of the earth and moon, the point o appears to be in the centre of the moon's disc, and the portion c o c' is visible, c r c' invisible. But when the observer is at a, the point p, on the right of o, appears in the centre, and the portion a p a' is visible, so that c' a' is now visible and a c invisible. In the same way, when the observer is at b, he can see the portion c b, while b' c' is invisible and q appears to be in the centre of the disc. Thus in the course of the day the portion a o b' (dotted in the figure) is constantly visible and b r a' (also dotted) constantly invisible, while a c b and a' c' b' alternately come into view and disappear. In other words, when the moon is rising we see a little more of the side which is the then uppermost, and when she is setting we see a little more of the other side which is uppermost in this position. A similar explanation applies when the observer is not on the earth's equator, but the geometry is slightly more complicated. In the same way, as the moon passes from south to north of the equator and back as she revolves round the earth, we see alternately more and less of the northern and southern half of the moon. This set of changes—the simplest of several somewhat similar ones which are now known as librations of the moon—being thus thought of as likely to occur, Galilei set to work to test their existence by observing certain markings of the moon usually visible near the edge, and at once detected alterations in their distance from the edge, which were in general accordance with his theoretical anticipations. A more precise inquiry was however interrupted by failing sight, culminating (at the end of 1636) in total blindness.
But the most important work of these years was the completion of the great book, in which he summed up and completed his discoveries in mechanics, Mathematical Discourses and Demonstrations concerning Two New Sciences, relating to Mechanics and to Local Motion. It was written in the form of a dialogue between the same three speakers who figured in the Dialogue on the Systems, but is distinctly inferior in literary merit to the earlier work. We have here no concern with a large part of the book, which deals with the conditions under which
