If, for example, in the figure, the circle represent a section of the sun by a plane through the observer at O, and A, B, C, D, E be points taken at equal distances along the surface of the sun, so as to represent the positions of an object on the sun at equal intervals of time, on the assumption that the sun revolves uniformly, then the apparent motion from A to B, as seen by the observer at O, is measured by the angle A O B, and is obviously much less than that from D to E, measured by the angle D O E, and consequently an object attached to the sun must appear to move more slowly from A to B, _i.e._ near the sun’s edge, than from D to E, near the centre. On the other hand, if the spot be a body revolving round the sun at some distance from it, _e.g._ along the dotted circle _c d e_, then if _c_, _d_, _e_ be taken at equal distances from one another, the apparent motion from _c_ to _d_, measured again by the angle _c_ O _d_, is only very slightly less than that from _d_ to _e_, measured by the angle _d_ O _e_. Moreover, it required only a simple calculation, performed by Galilei in several cases, to express these results in a numerical shape, and so to infer from the actual observations that the spots could not be more than a very moderate distance from the sun. The only escape from this conclusion was by the assumption that the spots, if they were bodies revolving round the sun, moved irregularly, in such a way as always to be moving fastest when they happened to be between the centre of the sun and the earth, whatever the earth’s position might be at the time, a procedure for which, on the one hand, no sort of reason could be given, and which, on the other, was entirely out of harmony with the uniformity to which mediæval astronomy clung so firmly. [Illustration: FIG. 56.—Galilei’s proof that sun-spots are not planets.] The rotation of the sun about an axis, thus established, might evidently have been used as an argument in support of the view that the earth also had such a motion, but, as far as I am aware, neither Galilei nor any contemporary noticed the analogy. Among other facts relating to the spots observed by Galilei were the greater darkness of the central parts, some of his drawings (see fig. 55) shewing, like most modern drawings, a fairly well-marked line of division between the central part (or =umbra=) and the less dark fringe (or =penumbra=) surrounding it; he noticed also that spots frequently appeared in groups, that the members of a group changed their positions relatively to one another, that individual spots changed their size and shape considerably during their lifetime, and that spots were usually most plentiful in two regions on each side of the sun’s equator, corresponding roughly to the tropics on our own globe, and were never seen far beyond these limits.