April 28, 1916] 



SCIENCE 



589 



symbols seized the most obvious means for 

 enlarging the scope and abbreviating the 

 processes of their favorite science! Into 

 the existing knowledge they brought order 

 and system, circumstances of the time gave 

 it rapid development, and a new branch of 

 science came into 'being. 



So projective geometry was cultivated. 

 It was the avowed rival of algebraic geom- 

 etry. The problems solved and new theories 

 advanced by Monge, Poncelet and Steiner 

 were matched by the genius of Pliicker, 

 Moebius, Cayley, Clifford, Cremona and 

 Sylvester. The theorems of the one kind, 

 resting on algebra, were perfectly general ; 

 those of the other, founded on intuition of 

 real elements, were compelled to state ex- 

 ceptions. To escape this obstacle, Poncelet 

 stated the postulate of continuity, a logical, 

 almost magical bridge over the lacunns. But 

 in algebra, when real quantities failed, 

 there were the imaginary quantities to fill 

 the gap. "What could pure geometry ex- 

 hibit as justification or explanation of the 

 continuity that she had postulated ? It was 

 a recluse professor in a provincial univei^ 

 sity, von Staudt, of Erlangen, who settled 

 the matter once for all with a perfect 

 analogy. As algebra defines imaginaries 

 by real quadratic equations whose roots are 

 not real, so, according to von Staudt, geom- 

 etry defines two imaginary elements by 

 two real pairs of elements. In certain 

 relative positions these determine two real 

 elements; otherwise they stand as a real 

 representation of two imaginaries. This is 

 genius: to define the required object by the 

 very phenomenon which constitutes the de- 

 mand. What is sauce for algebra is sauce 

 for geometry, and the imaginary elements 

 are since that time the secure possession of 

 both. 



What then were the conquests of alge- 

 braic geometry? The ancients had exam- 

 ined conies and conieoids, that is, circles, 



ellipses, spheres, ellipsoids, and those allur- 

 ing surfaces, the paraboloids, all loci of the 

 second order. Sir Isaac Newton had made 

 a pioneer study of plane curves of the 

 third order, a venture in which for more 

 than a century only two had followed him, 

 until Moebius and Pluecker, about 1835, re- 

 sumed the attack. It would take many 

 hours to name in most concise form the new 

 features and new problems that arose from 

 this study. Inflexional points, Steinerian 

 correspondences, poloconics, harmonic 

 polars, Cayleyan and Hessian covariant 

 curves — these will serve to remind some of 

 you of the multiple ramifications of in- 

 quiries that began on plane cubic curves. 

 Others -will recall the metrical properties 

 of semicubical parabolas and cissoids. Of 

 quartic curves, the next higher order, even 

 more is to be said — or omitted; their 28 

 double tangents and 24 inflexional points 

 and many seemingly elementary problems 

 connected with them remain unfinished, as 

 students in all lands can testify, among 

 others not a few Americans who have given 

 labor and time to them. 



Progress is often along converging lines. 

 While geometry advanced steadily in the 

 algebraic direction, algebra was acquiring 

 a new concept, that of a group of opera- 

 tions. Any set of operations form a group, 

 when two of them unite to form always a 

 third in the same set ; thus, uniform expan- 

 sions and contractions of an object form a 

 group, and in numbers all multiplications 

 and divisions together form a group. Now 

 a group of operations will change some 

 things and leave others unchanged or in- 

 variant. In algebra, the group of linear 

 substitutions was the first to attract atten- 

 tion, and between 1845 and 1865 the study 

 of this group was the most conspicuous 

 business of algebraists. Soon it was recog- 

 nized that in geometry all projective trans- 

 formations constitute a group, and that 



