186. MAGNITUDES.--The telescopic stars show among themselves an even greater range of brightness than do the lucid ones, and the system of magnitudes (§ 9) has accordingly been extended to include them, the faintest star visible in the greatest telescope of the present time being of the sixteenth or seventeenth magnitude, while, as we have already learned, stars on the dividing line between the telescopic and the lucid ones are of the sixth magnitude. To compare the amount of light received from the stars with that from the planets, and particularly from the sun and moon, it has been found necessary to prolong the scale of magnitudes backward into the negative numbers, and we speak of the sun as having a stellar magnitude represented by the number -26.5. The full moon's stellar magnitude is -12, and the planets range from -3 (Venus) to +8 (Neptune). Even a very few of the stars are so bright that negative magnitudes must be used to represent their true relation to the fainter ones. Sirius, for example, the brightest of the fixed stars, is of the -1 magnitude, and such stars as Arcturus and Vega are of the 0 magnitude. 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 100^{1/5} = 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: