The Hindu philosophers reckon seven magnitudes as follows:—

The Greeks mention six. The first they call the greatest (Akbar) and the sixth, the least (Asghar), and each comprised three degrees, the great, the mean and the less, each more important in proportion to its degree.* The intervals of the hexade were measured by sixths. Some supposed that a diameter of a star of the 1st magnitude was six times the diameter of the smallest; but a manifest error occurred in calculating the volumes and distances intervening, by concluding that the volume of a mean star of the 1st magnitude must therefore be six times larger than the volume of a star of the 6th magnitude. But Euclid has demonstrated in the last proposition of the 12th Book of the Elements, that circles are to one another as the squares on their diameters, that is, if the ratio of one diameter to another be one-half or less, there will be three times the ratio between the spheres. For instance, if the diameter of one sphere be half the diameter of another, the smaller sphere will be ½ of ½ of ½ or ⅛ of the larger; and if the diameter be ⅓, the smaller sphere will be ⅓ of ⅓ of ⅓ or ½7 of the larger, and so on. Therefore, if the case be as those have con­jectured, the volume of a star of the 1st magnitude will be greater than that of one of the 6th by a very considerable difference.

The largest of the fixed stars that have been observed, is 222 times, and the smallest of them twenty-three times as large as the earth. From their multitude they cannot be numbered, but the position of 1022* has been fixed. Of these—

There are besides, 14 whose magnitudes are not catalogued, nine of which are obscure and five nebular. This is the theory of Ptolemy. According to Abdúl Raḥmán-b-Omar al Ṣúfi,’*