The relation of these magnitudes to each other has been so chosen that a star of any one magnitude is very approximately 2.5 times as bright as one of the next fainter magnitude, and this ratio furnishes a convenient method of comparing the amount of light received from different stars. Thus the brightness of Venus is 2.5 × 2.5 times that of Sirius. The full moon is (2.5)9 times as bright as Venus, etc.; only it should be observed that the number 2.5 is not exactly the value of the light ratio between two consecutive magnitudes. Strictly this ratio is the 5√ 100 = 2.5119+, so that to be entirely accurate we must say that a difference of five magnitudes gives a hundredfold difference of brightness. In mathematical symbols, if B represents the ratio of brightness (quantity of light) of two stars whose magnitudes are m and n, then
B = (100)(m-n)/5
How much brighter is an ordinary first-magnitude star, such as Aldebaran or Spica, than a star just visible to the naked eye? How many of the faintest stars visible in a great telescope would be required to make one star just visible to the unaided eye? How many full moons must be put in the sky in order to give an illumination as bright as daylight? How large a part of the visible hemisphere would they occupy?
187. Classification by magnitudes.—The brightness of all the lucid stars has been carefully measured with an instrument (photometer) designed for that special purpose, and the following table shows, according to the Harvard Photometry, the number of stars in the whole sky, from pole to pole, which are brighter than the several magnitudes named in the table:
| 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 × 3m | 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.
The magnitudes of the telescopic stars have not yet been measured completely, and their exact number is unknown; but if we apply our principle of a threefold increase for each successive magnitude, we shall find for the fainter stars—those of the tenth and twelfth magnitudes—prodigious numbers which run up into the millions, and even these are probably too small, since down to the ninth or tenth magnitude it is certain that the number of the telescopic stars increases from magnitude to magnitude in more than a threefold ratio. This is balanced in some degree by the less rapid increase which is known to exist in magnitudes still fainter; and applying our formula without regard to these variations in the rate of increase, we obtain as a rude approximation to the total number of stars down to the fifteenth magnitude, 86,000,000. The Herschels, father and son, actually counted the number of stars visible in nearly 8,000 sample regions of the sky, and, inferring the character of the whole sky from these samples, we find it to contain 58,500,000 stars; but the magnitude of the faintest star visible in their telescope, and included in their count, is rather uncertain.