The number of stars brighter than magnitude 1.0 is 11 " " " " " 2.0 " 39 " " " " " 3.0 " 142 " " " " " 4.0 " 463 " " " " " 5.0 " 1,483 " " " " " 6.0 " 4,326 It must not be inferred from this table that there are in the whole sky only 4,326 stars visible to the naked eye. The actual number is probably 50 or 60 per cent greater than this, and the normal human eye sees stars as faint as the magnitude 6.4 or 6.5, the discordance between this number and the previous statement, that the sixth magnitude is the limit of the naked-eye vision, having been introduced in the attempt to make precise and accurate a classification into magnitudes which was at first only rough and approximate. This same striving after accuracy leads to the introduction of fractional numbers to represent gradations of brightness intermediate between whole magnitudes. Thus of the 2,843 stars included between the fifth and sixth magnitudes a certain proportion are said to be of the 5.1 magnitude, 5.2 magnitude, and so on to the 5.9 magnitude, even hundredths of a magnitude being sometimes employed. We have found the number of stars included between the fifth and sixth magnitudes by subtracting from the last number of the preceding table the number immediately preceding it, and similarly we may find the number included between each other pair of consecutive magnitudes, as follows: Magnitude 0 1 2 3 4 5 6 Number of stars 11 28 103 321 1,020 2,843 4 × 3^{m} 12 36 108 324 972 2,916 In the last line each number after the first is found by multiplying the preceding one by 3, and the approximate agreement of each such number with that printed above it shows that on the whole, as far as the table goes, the fainter stars are approximately three times as numerous as those a magnitude brighter.