Such photographs of the stars are now constantly reproduced in works on Astronomy and in popular magazine articles, and although some of them are very striking, many persons are disappointed with them, and cannot understand their great value, because each star is represented by a white circle often of considerable size and with a somewhat undefined outline, not by a minute point of light as stars appear in a good telescope. But the essential matter in all such photographs is not so much the smallness, as the roundness, of the star-images, as this proves the extreme precision with which the image of every star has been kept by the clockwork motion of the instrument on the same point of the plate during the whole exposure. For example, in the fine photograph of the Great Nebula in Andromeda, taken 29th December 1888, by Dr. Isaac Roberts, with an exposure of four hours, there are probably over a thousand stars large and small to be seen, every one represented by an almost exactly circular white dot of a size dependent on the magnitude of the star. These round dots can be bisected by the cross hairs of a micrometer with very great accuracy, and thus the distance between the centres of any of the pairs, as well as the direction of the line joining their centres, can be determined as accurately as if each was represented by a point only. But as a minute white speck would be almost invisible on the maps, and would convey no information as to the approximate magnitude of the star, mistakes would be much more easily made, and it would probably be found necessary to surround each star with a circle to indicate its magnitude, and to enable it to be easily seen. It is probable, therefore, that the supposed defect is really an important advantage. The above-mentioned photograph is beautifully reproduced in Proctor's _Old and New Astronomy_, published after his greatly lamented death. But besides the amount of altogether new knowledge obtained by the methods of research here briefly explained, a great deal of light has been thrown on the distribution of the stars as a whole, and hence on the nature and extent of the stellar universe, by a careful study of the materials obtained by the old methods, and by the application of the doctrine of probabilities to the observed facts. In this way alone some very striking results have been reached, and these have been supported and strengthened by the newer methods, and also by the use of new instruments in the measurement of stellar distances. Some of these results bear so closely and directly upon the special subject of the present volume, that our next chapter must be devoted to a consideration of them. CHAPTER IV THE DISTRIBUTION OF THE STARS