March 10, 1911] 



SCIENCE 



359 



by starting a discussion of imaginary num- 

 bers, liypdnsfjaco, or the iion-cuciidtian 

 geometries. It is easy to find men who 

 will mark out the regions within which 

 mathematicians should exercise their pow- 

 ers. It is an interesting psychological 

 phenomenon that a specialist who has 

 spent many years on a subject and has 

 become a recognized authority in it seldom, 

 if ever, will make any definite and general 

 statement in regard to it; yet often he will 

 not hesitate to make sweeping dogmatic 

 assertions respecting things entirely out- 

 side his line, for example (to use a harm- 

 less illustration), respecting the merits of 

 the tariff or the crime of seventy-three. 

 Even those who have been expert in math- 

 ematics have differed much among them- 

 selves respecting what should be its highest 

 aims. Fourier, in reporting on the work 

 of Jacobi to the Academy of Sciences, said 

 that natural philosophy should be the prin- 

 cipal object of the meditations of jnatlie- 

 maticians. In the introduction to his the- 

 ory of heat referring to analysis he wrote 

 "there could not be a language more 

 universal and more simple, more exempt 

 from errors and obscurities, that is to say, 

 more worthy of expressing the invariable 

 relations of natural objects. Considered 

 from this point of view, it is coextensive 

 with nature itself; it defines all the sen- 

 sible relations, measures the times, the 

 spaces, the forces, the temperatures; this 

 difficult science is formed slowly, but it 

 retains all the principles it has once ac- 

 quired. It grows and becomes more cer- 

 tain without limit in the midst of so many 

 errors of the human mind." Replying to 

 the reproach of P\jurier, Jacobi, in a letter 

 to Legendre, said: "It is true that M. 

 Fourier had the opinion that the principal 

 end of mathematics was the public utility 

 and the explanation of natural phenom- 

 ena; but such a philosopher as he is should 



have known that the unique end of science 

 is the honor of the human mind, and that 

 from this point of view a qiiestion of num- 

 ber is as important as a ([uestion of the 

 system of the world." (jiauss agreed, for 

 he said that mathematics is the queen of 

 the sciences, and that arithmetic is the 

 queen of mathematics. 



Obviously it is just that th(! astronomer 

 should allow the mathematician all the 

 latitude in defining the limits of mathe- 

 matics that he himself would desire if he 

 were permitted to mark out the borders of 

 the field of astronomy. It would be con- 

 sidered unwarrantable interference and an 

 evidence of hopeless ignorance if any group 

 of men should attempt to make astron- 

 omers confine themselves to those phases of 

 their subject which are immediately useful 

 to a busy world. If astronomy were limited 

 simply to those f)arts which are necessary 

 for time service on the land and the use of 

 navigators on the sea ; if it were necessary to 

 abandon those mathematical thoori(!S of the 

 motions of the planets and satellites which 

 are in all respects the most perfect ex- 

 amples in natural science of harmony be- 

 tween theory and observation ; if it were no 

 longer permitted to use our powerful in- 

 struments in observing the peculiarities of 

 the planets and the sun; if we were com- 

 pelled to discontinue our investigation 

 as to their origin and evolution; if we 

 were under obligations to give up the 

 spectroscope forever; and if we were forced 

 to forego all further attempts to sound the 

 almost boundless depths of the sidereal 

 system and unravel its mysteries; if astron- 

 omy were put under these restraints, I say, 

 then most of those incentives which in all 

 the history of astronomical science have 

 produced the rarest examples of devotion 

 to ideals and the pursuit of knowledge 

 would be removed. Astronomers do not 

 admit the right of a jtartially informed 



