MATHEMATICS IN THE NINETEENTH CENTURY 487 



use of the abbreviate notation which permits us to study the proper- 

 ties of geometric figures without the intervention of the coordinates, 

 the introduction of line and plane coordinates, and the notion of 

 generalized space elements. Steiner, who has been called the greatest 

 geometer since Appolonius, besides enriching geometry in countless 

 ways, was the first to employ systematically the method of generating 

 geometrical figures by means of protective pencils. 



Other noteworthy works belonging to this period are Pliicker's 

 System der analytischen Geometric (1835), and Chasles's classic Apercu 

 (1837). 



Already at this stage we notice a bifurcation in geometrical 

 methods. Steiner and Cha'sles become eloquent champions of the 

 synthetic school of geometry, while Pliicker, and later Hesse and 

 Cayley, are leaders in the analytical movement. The astonishing 

 fruitfulness and beauty of synthetic methods threatened for a short 

 time to drive the analytic school out of existence. The tendency 

 of the synthetic school was to banish more and more metrical methods. 

 In effecting this the anharmonic ratio became constantly more promi- 

 nent. To define this fundamental ratio without reference to measure- 

 ment, and so free projective geometry from the galling bondage 

 of metric relations, was thus a problem of fundamental importance. 

 The glory of this achievement, which has, as we shall see, a far 

 wider significance, belongs to v. Staudt. Another equally important 

 contribution of v. Staudt to synthetic geometry is his theory of 

 imaginaries. Poncelet, Steiner, Chasles operate with imaginary 

 elements as if they were real. Their only justification is recourse to 

 the so-called principles of continuity or to some other equally vague 

 principle. V. Staudt gives this theory a rigorous foundation, defining 

 the imaginary points, lines, and planes by means of involutions 

 without ordinal elements. 



The next great advance made is the advent of the theory of alge- 

 braic invariants. Since projective geometry is the study of those 

 properties of geometric figures which remain unaltered by projective 

 transformations, and since the theory of invariants is the study of 

 those forms which remain unaltered (except possibly for a numerical 

 factor) by the group of linear substitutions, these two subjects are 

 inseparably related and in many respects only different aspects of the 

 same thing. It is no wonder, then, that geometers speedily applied 

 the new theory of invariants to geometrical problems. Among the 

 pioneers in this direction were Cayley, Salmon. Aronhold, Hesse, 

 and especially Clebsch. 



Finally we must mention the introduction of the line as a space 

 element. Forerunners are Grassmann (1844) and Cayley (1859), but 

 Pliicker in his memoirs of 1865, and his work Neue Geometric des 

 Raumes (1868-69), was the first to show its great value by studying 



