Januaey 19, 1900.] 



SCIENCE. 



83 



The development of this branch of science 

 along with the development of the closely 

 related science of geodesy, is a work essen- 

 tially of the present century, and must be 

 attributed chiefly to the German school of 

 astronomers led by Gauss and Bessel. It 

 is to these eminent minds, as well known 

 in pure as in applied mathematics, that we 

 are indebted for the theories, and for the 

 most advantageous methods of use, of in- 

 strumental appliances, and for the refined' 

 processes of numerical calculation which 

 secure the best results from observational 

 data. It is a fortunate circumstance, per- 

 haps, considering the irreverence which 

 some modern pure mathematicians show 

 for numerical computations, that Gauss and 

 Bessel began their careers long before the 

 resistless advent of the theory of functions 

 and the theory of groups. 



The story of the opportune discovery of 

 the planet Ceres, as related by Gauss him- 

 self in the preface to his Theoria Motus 

 Corporum Ccelestium, is well known ; but 

 it is less well known that the merit of this 

 magnificent work lies rather in the model 

 groups of formulas presented for the precise 

 numerical solution of intricate problems 

 than in the facility afforded for locating the 

 more obscure members of the solar system. 

 Indeed, the works of Gauss and Bessel are 

 everywhere characterized by a clear recog- 

 nition of the important distinction between 

 those solutions of problems which are, and 

 those which are not, adapted to numerical 

 calculation. They showed astronomers how 

 to systematize, to expedite, and to verify 

 arithmetical operations in ways which were 

 alone adequate to the accomplishment of 

 the vast undertakings which have since 

 been completed in mathematical geodesy 

 and in sidereal astronomy. 



Among the most important contributions 

 of these authors to geodesy and astronomy 

 in particular, and to the precise observa- 

 tional sciences in general, is that branch 



of the theory of probability called the 

 ' method of least squares.' * No single ad- 

 junct has done so much as this to perfect 

 plans of observation, to systematize schemes 

 of reduction, and to give definiteness to 

 computed results. The effect of the general 

 adoption of this method has been somewhat 

 like the effect of the general adoption by 

 scientific men of the metric system ; it has 

 furnished common modes of procedure, 

 common measures of precision, and com- 

 mon terminology, thus increasing to an 

 untold extent the availability of the price- 

 less treasures which have been recorded in 

 the century's annals of astronomy and 



When we pass from the field of observa- 

 tional astronomy to the more restricted but 

 more intricate field of dynamical astron- 

 omy, it is apparent that Laplace and his 

 contemporaries quite underestimated the 

 magnitudes of the mathematical tasks they 

 bequeathed to their successors. Laplace, 

 almost unaided, had performed the un- 

 paralleled feat of laying down a complete 

 outline of the ' system of the world ' ; but 

 the labor of filling in the details of that 

 outline, of bringing everj' member of the 

 solar system into harmony at once with the 

 simple law of gravitation and with the 

 inexorable facts of observation, is a still 

 greater feat which has taxed the combined 

 eflbrts of the most acute analysts and the 

 most skillful computers of the preceding 

 and present generation. 



It is impossible within the limits of a 

 semi-popular address to do more than men- 

 tion in the most summary way the extraor- 

 dinary contributions to dynamical astron- 

 omy made especially during the present 



* Gauss's fundamental paper in this subject is 

 "Theoria combinationis observationum erroribus 

 minimis obnoxite, " and dates from 1821. Werke, 

 Band IV. 



Bessel's numerous contributions to this subject are 

 fojind in his " Abhandlungen" cited above. 



