What the work gives is, in the first place, the magnitude and approximate position of every star photographed. The determining of the magnitude of a star is an important and delicate question. There is no difficulty in determining, from the diameter of the image of the star as seen in the microscope, what its photographic magnitude was at the time of the exposure, as compared with other stars on the same plate. But can we rely upon similar photographic magnitudes on a plate corresponding to similar brightnesses of the stars? In the opinion of Gill and Kapetyn we cannot. The transparency of the air varies from night to night, and on a very clear night the same star will give a stronger image than it will when the air is thick. Besides, slightly different instruments were used in the course of the work. For these reasons a scale of magnitude was determined on each plate by comparing the photographic intensity of the images of a number of stars with the magnitudes as observed with the eye by various observers. Thus on each plate the magnitude was reduced to a visual scale. It does not follow from this that the magnitudes are visual, and not photographic. It is still true that a blue star will give a much stronger photographic image than a red star of equal visual brightness. In a general way, it may be said that the catalogue includes all the stars to very nearly the tenth magnitude, and on most of the plates stars of 10.5 were included. In fact, now and then is found a star of the eleventh magnitude. A feature of the work which adds greatly to its value is a careful and exhaustive comparison of its results with previous catalogues of the stars. When a star is found in any other catalogue the latter is indicated. Most interesting is a complete list of catalogued stars which ought to be on the photographic negatives, but were not found there. Every such case was inexhaustibly investigated. Sometimes the star was variable, sometimes it was so red in color that it failed to impress itself on the plate, sometimes there were errors in the catalogue. The great enterprise of making a photographic map of the heavens now being carried on as an international enterprise, having its headquarters at Paris, is yet wider in its scope than the works we have just described. One point of difference is that it is intended to include all the stars, however faint, that admit of being photographed with the instruments in use. The latter are constructed on a uniform plan, the aperture of each being 34 centimetres, or 13.4 inches, and the focal length 343 cm. Two sets of plates are taken, one to include all the stars that the instrument will photograph near poles, and the other only to take in those to the eleventh magnitude. Of the latter it is intended to prepare a catalogue. Some portions of the German and English catalogues have already been published, and their results will be made use of in the course of the present work. NUMBERING THE STARS.