Apeil 28, 1916] 



SCIENCE' 



587 



complexity. It is not surprising therefore 

 to find Gibbs almost equally distinguished 

 in difficult fields of pure mathematics, the 

 geometry of iV dimensions and in vector 

 algebra. 



I have not yet mentioned astronomy, nor 

 living scientists ; but can not forbear to call 

 your attention to the apparent decline and 

 fall of the Laplacean theory of the earth's 

 genesis from a nebula, its slow concentration 

 and shrinkage. Two eminent scientists of 

 the present day. Chamberlain and Moulton, 

 have resolutely insisted upon precise formu- 

 lation of hypotheses, have subjected them 

 to calculation as exact as the case admits, 

 and seem to have established the superior 

 probability of their planetesimal hypothesis : 

 that the major part at least of the earth's 

 mass is the result of slow accretions from in- 

 tercepted streams of meteorites. It may be 

 that a new theory of nebular evolution must 

 be constructed, starting from the spiral ar- 

 rangement visible in so many of Hale's and 

 Eitehie's photographs. Certainly there is 

 a strong temptation for younger scientists 

 to join in the working out of this great 

 problem, now successfully past its initial 

 stage. 



Most scientists can and wiU become 

 mathematicians when their special problems 

 reach the stage where measurements are 

 possible, and pure mathematicians should 

 be eager to discuss concrete problems when 

 they see the possibility of applying meth- 

 ods that they undei^stand. But there is an 

 independent territory of pure mathematics, 

 a realm of the understanding and the rea- 

 son. Its fields have been explored, subdued 

 and cultivated bj men of genius, men of 

 strong imagination, men of patient dili- 

 gence, and by adventurers, ever since the 

 beginnings of history. If the triumphs of 

 natural science loom larger before the eyes 

 of the average man, it is on account of his 

 intellectual position and the distortions of 



perspective. Has man yet included in his 

 scientific knowledge such a part of the now 

 knowable universe as would be represented 

 by any finite fraction, however small? Is 

 all that the whole race has known, com- 

 pared with the secrets yet to be discovered, 

 as considerable as the smallest twinkling 

 star among the fiery millions of the galaxy ? 

 Are not aU scientists, with Sir Isaac New- 

 ton, children wandering beside the sea and 

 gathering the pebbles that please them? 

 The intellectual booty gathered by pure 

 mathematicians in the past century was 

 relatively not less magnificent than the 

 fragments of understanding secured by 

 scientists of the concrete; nor is the one 

 kind, in the last analysis, more purely intel- 

 lectual than the other. All true science is 

 rational, and belongs equally to the reason. 

 I shall name but few of the nineteenth cen- 

 tury discoveries in pure mathematics, and 

 not, perhaps, the most important; for in 

 the record of times so recent each wUl nec- 

 essarily praise the things that he himself 

 has most admired. 



Geometry has made more numerous and 

 more important advances than in the previ- 

 ous three centuries. Her devotees have 

 been numbered by the hundreds. Read the 

 appreciative chronicles of Professor Gino 

 Loria in "II passato ed il presente delle 

 prineipali teorie geometriche, " in which he 

 sets forth a noteworthy thesis, first pro- 

 pounded by the great French geometrician, 

 Chasles. This science, he declares, is the 

 most attractive, because the humblest 

 worker may hope by diligence not merely to 

 survey the edifice, but to build it further. 

 Genius is no longer indispensable to him 

 who would add a stone to its walls. This 

 exhortation is equally valid to-day; and 

 for this reason a young mathematician may 

 well devote some time to geometry, even if 

 his ultimate dream leads elsewhere. 



The earliest years of the century saw the 



