Amongst the brighter stars there is a close agreement in the estimate of magnitude by different observers, but amongst the higher magnitudes a difference appears. Sir J. Herschel and Admiral Smyth, for instance, go into much higher numbers of magnitudes than Struve; the limit of Admiral Smyth’s vision with his 6-inch telescope was a 16th magnitude, while the limit of Struve’s vision with a 9½-inch telescope he calls a 12th magnitude; the estimates of the latter observer are, however, gaining greater adoption. In order to reduce the relative magnitude to a law, Mr. Pogson[21] took stars differing largely in magnitude, and compared the amount of light from each, and so reduced the ratio between the magnitudes given by Knott and all the best observers. From this he found that a mean of 2·4 represented the ratio, and for reasons given he adopted the quantity 2·512 as a convenient ratio; as he states, “the reciprocal of ½ log. R (in his paper R = the ratio 2·512), a constant continually occurring in photometric formulæ, is in this case exactly 5.” So far the ratio is established. The next thing is the basis from which to commence reckoning; this Mr. Pogson fixed by reference to Argelander’s catalogued stars, estimated by him at about the 9th magnitude, and with these, comparison is made with the star whose light is measured, and the above constant of ratio applied, which at once gives the magnitude of the measured star. To do this, in Mr. Pogson’s words: “If then any observer will determine for himself the smallest of Argelander’s magnitudes, just visible by fits, on a fine moonless night, with an aperture of one inch, and call this quantity L, or the limit of vision for one inch, the limit _l_, for any other aperture, will be given by the simple formula, _l_ = L + 5 × log. aperture.” The value of L founded by Mr. Pogson is 9·2; that is, a star of 9·2 magnitude, according to Argelander, is limited by 1-inch aperture, with Mr. Pogson’s eye. On different nights and with different eyes, this number, or the magnitude limited, must vary, and it varies from exactly the same causes that produce variation in the light of the stars to be measured, so that we are independent of transparency of the air, at least within considerable limits. Having found the value of L for any night, we turn the telescope on a star to be measured, then alter the aperture if we employ the first method, until the limit is found, and insert the value in the equation, the value of _l_, or the star’s magnitude, then at once appears. By this means a number of well-known stars of all magnitudes may be settled for future reference and comparison with variable stars.