In all work of this kind the observer must look directly at the star he is observing at the moment, and never try to compare two stars by a simultaneous inspection of both. After examining one star until he has a distinct impression of its average brightness, freed from the momentary changes due to atmospheric disturbance, he should observe the other in the same manner. Alternate observations of the two stars, each observation lasting for a few seconds, will give a truer impression than can be derived from a simultaneous observation in which the two images must be differently placed on the retina. The principal objection to this method is the difficulty of determining the value of a grade, as it is liable to vary with the observer, the time, the condition of the air, and the brightness of the stars. These difficulties are avoided by the following method. Select two stars for comparison; one, _a_, slightly brighter than the star to be measured, _v_, the other, _b_, slightly fainter. The interval between _a_ and _b_ should never exceed one magnitude. Estimate the brightness of _v_ in tenths of the interval from _a_ to _b_. Thus, if _v_ is midway between _a_ and _b_ the interval will be five tenths, and we may write _a_ 5 _b_. If _v_ is nearly as bright as _a_, we may have _a_ 1 _b_ or _a_ 2 _b_; if _v_ is not much brighter than _b_, we may have _a_ 8 _b_ or _a_ 9 _b_. An advantage of this method is that larger intervals in brightness may be used between the comparison stars, and accordingly less distant stars employed. An increase in distance of the stars always renders the comparison more difficult. We can also obtain many independent comparisons by using several comparison stars. If we have _m_ stars brighter and _n_ fainter, we shall only have _m_ + _n_ independent measures by the method of grades, while we may have _m n_ comparisons by estimating tenths, since estimates may be made in terms of the intervals between each brighter and each fainter star. On the other hand, especially when observing stars not very near together, it is a decided advantage to have to compare two stars rather than three. Each method has its advantages, and that to be used should doubtless depend on the temperament of the observer.