110 



SCIENCE 



[N. S. Vol. XLII. No. 1073 



In 1868 and 1869 was published an 

 epoch-making work, the "Geometry of 

 Straight Lines," by Julius Pliieker, pro- 

 fessor of physics and mathematics at the 

 University of Bonn. Unconsciously stu- 

 dents of geometry had presumed that the 

 space we live in must be conceived as built 

 up of points, or minute bodies, having 

 vanishing magnitude or dimension in all 

 directions. Pliieker pointed out that to the 

 eye of pure mathematics it is just as true 

 that space is built of straight lines. The 

 intersection of the lines of geometry is 

 no obstacle to such a theory, any more than 

 if they were rays of starlight! With this 

 new conception of space, amenable to a 

 beautiful algebraic treatment, came a great 

 stimulation of speculation and exploration 

 in geometrical realms both old and new. 

 Pliieker, his brilliant pupil Klein, Clifford 

 and Cayley are the centers of greatest 

 energy in this movement, whose force per- 

 sisted for twenty years and more. I show 

 here a graph for geometry, no longer as a 



Fig. 7. Geometry, Number of Titles; Number 

 of Pages, for Comparison, sliown by Dotted Curve. 

 Agreement, fair. 



greater or less part of the entire range of 

 mathematics, but measured, so to say, abso- 

 lutely — the number of titles or of pages 

 given to it in each annual volume of the 

 Jahrbuch. Judged upon either basis, 

 geometry seems to have doubled in rate of 

 production from 1870 to 1890, then to have 

 fallen off a third, to regain most of this loss 

 after 1899. 



In the universities of Germany the de- 



cline was decidedly more violent than this 

 graph can show; for men trained in a 

 specialty do not all change suddenly their 

 interest or their line of study. So the new 

 impulse, if it comes, will lag in its mani- 

 festation on this graph. Geometry was 

 transplanted into Italy during the '80 's, 

 and the notion of Group, worked out in 

 partial applications earlier by Klein, 

 Sophus Lie, and their schools, was extended, 

 energized and developed in larger geometric 

 shape by young, ardent, brilliant sons of 

 the men who in 1870 had created a new 

 kingdom of Italy. A new society began to 

 publish Bendiconti at Palermo ('88) and 

 we note the point, about 1893, where this 

 rising tide meets and overcomes the ebb of 

 the German wave, Segre is here the great 

 name, at first; Castelnuovo and Enriques 

 soon rise to altitudes before unknown, 

 algebraic surfaces now proving to be no less 

 interesting than algebraic curves had been 

 in the 70 's and 80 's. 



Tig. 8. Above, Analytic Geometry, Titles; Be- 

 low, Modern Synthetic Geometry (same scale). 



Let us further disting-uish, dividing 

 geometry into the pure or synthetic, and the 

 algebraic. Both had experienced revival in 

 Gennany, the synthetic geometry through 

 a combination of influences, conspicuously 

 through the publication of Reye's lectures 

 in form pedagogically perfect, with full 

 illustration and touched with that scientific 

 fire which is almost poetic inspiration. 

 Our graph is arranged to show on opposite 

 sides of a base-line, by numbers of titles, 

 the output of analytic and algebraic geome- 

 try above, the synthetic below. The ordi- 

 nates are absolute, not percentages; and 

 further argument is unnecessary to estab- 



