Maech 10, 1911] 



SCIENCE 



361 



later in connection with the solution of the 

 same problems. 



We shall not, however, get any compre- 

 hensive view of the relations of astronomy 

 and mathematics by citing, without some 

 classification, isolated examples where the 

 latter is indebted to the former. Such a 

 procedure will give us no idea of the 

 reasons for any of the great movements in 

 mathematical thought. Moreover, the 

 mathematical theories are so interwoven 

 that it is difficult to pick out individual 

 branches and to discuss their origins with- 

 out being at least very incomplete. There- 

 fore we shall content ourselves with broad 

 classifications of mathematics, and to state- 

 ments, with illustrations, of those parts 

 where the practical problems of astronomy 

 have had important influences. 



Mathematics may first be divided into 

 the metrical and the non-metrical branches. 

 The former are vastly more important 

 than the latter ; or, since it is perhaps wise 

 to avoid passing judgment as to what is 

 important, the metrical branches have at 

 least an enormously greater literature than 

 the non-metrical. Recognizing the fact 

 that there is much of a non-metrical char- 

 acter in those subjects which are regarded 

 as being on the whole metrical in nature, 

 and not wishing to insist on the possibility 

 of making an absolute division on this 

 principle, an examination of the Royal So- 

 ciety Index covering pviblications in mathe- 

 matics from 1800 to 1900 shows that prob- 

 ably not one part in forty has been de- 

 voted to non-metrical mathematics. While 

 there are certain non-metrical aspects of 

 some astronomical problems, it would not 

 be fair to claim that astronomy has had any 

 essential part in inspiring these branches 

 of mathematics. 



Now considering only the metrical 

 branches of mathematics, we may divide 

 them into the mathematics of the continu- 



ous and the mathematics of the discrete. 

 Here the problem of actually effecting the 

 division is even more difficult than in the 

 preceding case, for there is more inter- 

 mingling and there seem to be more de- 

 batable points. In spite of these difficulties 

 the mode of division is attached to certain 

 fundamental characteristics either of the 

 subject matter or of the processes employed. 

 The ordinary theory of numbers is an ex- 

 ample of the mathematics of the discrete. 

 The theory of ordinary equations, for ex- 

 ample, linear equations, may be considered 

 as an example of the mathematics of the 

 discrete or the continuous, according as the 

 coefficients are regarded as discrete niun- 

 bers or continuous functions of certain 

 parameters. In such cases where the ideas 

 of continuity are not essential to the formu- 

 lation and treatment of the problems, they 

 will be considered as belonging to the 

 mathematics of the discrete. All those 

 branches of mathematics in which contin- 

 uity is an essential feature, as, for example, 

 those involving derivatives, constitute the 

 mathematics of the continuous. 



On the whole, the problems of astron- 

 omy have not given rise to the mathematics 

 of the discrete. While the physical uni- 

 verse seems to be made out of discrete 

 things — atoms, corpuscles, units of elec- 

 tricity — it changes continuously from one 

 state to another. Since a large part of the 

 problems of the natural sciences relate to 

 changes of position or state, such as the 

 motion of a world or the evolution of an 

 animal, this continuity is forced into the 

 foreground in the applications of mathe- 

 matics to physical questions. Consequently, 

 in seeking for places where astronomy has 

 made real contributions to mathematical 

 theory we may restrict our search to the 

 mathematics of the continuous. If there 

 are no other subjects which have made 

 similar contributions, we have at once the 



