GRAVITATION BROWN 185 



exactly. What is actually done is the introduction of modifications 

 made to simplify the mathematical and numerical work; the results 

 so obtained are later corrected to take care, as far as possible, of 

 the changes that have been introduced. This process of gradual 

 approximation is in general use for most of the problems in which 

 mathematical methods are adopted for the study of physical phe- 

 nomena; for those of celestial mechanics, it is the basis of all the 

 methods which have been devised. 



After Laplace's time the increasing accuracy and number of the 

 observations demanded that the calculations giving the positions of 

 the moon and planets be carried out in much greater detail, involving 

 a disproportionate increase in the work. The problem of the moon's 

 motion alone required many years of calculation to carry the ap- 

 proximations to the point where the results appeared to have the 

 same degree of accuracy as the observations. During the nineteenth 

 century only two of the many who started succeeded in getting near 

 the goal, namely, P. A. Hansen and C. Delaunay. Hansen's w^ork, 

 with minor corrections, w^as used to calculate the positions of the 

 moon given in the Nautical Almanacs for some TO years. Delaunay's 

 theory was not put into tabular form until the beginning of the 

 present century, at a time when the demand for accuracy exceeded 

 that which his theory was capable of furnishing. 



For the great planets two ambitious projects have been carried 

 out successfully. In Paris, Leverrier, w4th the help of a staff of 

 computers, recalculated the theories of their motions and carried the 

 work through to the formation of tables which were in use for many 

 years to predict their places given in the almanacs. Toward the close 

 of the century, a similar task was undertaken by Simon Newcomb 

 in Washington, but with increased demands for accuracy ho found 

 the work too great for one man, even with the computing assistance 

 which he was able to command. He wisely sought and obtained the 

 services of (x. W. Hill, the greatest celestial mechanician so far arisen 

 in America, for the most laborious and difficult part of the work, 

 namely, the theories of the motions of Jupiter and Saturn. 



With the publication of the works of Hill and Newcomb carried 

 through to the formation of tables so that comparison with observa- 

 tion was rendered comparatively easy, it w^as evident that the accu- 

 racy of the theories had caught up with that of the observations. It 

 thus became possible to discuss, with the hope of making new dis- 

 coveries, the differences between the theories and the immense store 

 of observations that had been accumulated at many observatories 

 during the nineteenth century. Newcomb undertook this task and 

 proved himself to be a master in marshaling great masses of mate- 

 rial and in deducing: the best results from them. The details of this 



