emch: projective groups. 33 



H e = CO 1 H ? 



§6. Historical Sketch. 



To show what position the subject of this dissertation occupies in 

 geometry it will be necessary to give first a brief account of the 

 development of geometrical methods which gradually lead to 

 the modern standpoint. Projective or synthetic geometry is 

 essentially a product of the 19th century, though it is well known 

 that Pappus and Menelaus found some very important theorems 

 concerning projective properties many hundred years ago and that 

 Desargues discovered the fundamental theorems of perspective 

 and involution in the i8th century. 



The origin of projective geometr}- must be sought in the methods 

 of descriptive geometr}', which, by the achievements of Lambert 

 and Monge, became at once very valuable in geometrical investi- 

 gation. The first classic work on projective geometry was Ponce- 

 let's "Traite des proprietes projectives des figures," which appeared 

 in 1822. In this great treatise the properties of figures are inves- 

 tigated which are unaltered b}^ projection, or which are invariant. 

 Poncelet introduced the so-called central-projection with a per- 

 spective-centre and a perspective-axis into the consideration of 

 plane figures. While in France the "new geometry" was chiefly 

 promoted by Gergonne and Chasles, in Germany its fruitfulness 

 was shown to the scientific world by the three great investigators, 

 Mobius, Pliicker, and Steiner. The classical works of this period 

 are: 



Mobius, Barjxentrischer Calcul, 1827. 



Pliicker, Analytisch-geometrische Untersuchungen, 1828. 

 Steiner, Systematische Entwickelung der Abhangigkeit geome- 



trischer Gestalten von einander, 1832. 

 Chasles, Apercu historique sur 1' origine et le developpement des 

 methodes en geometrie, 1838. 



From these times also dates the separation of the mathematicians 

 into two schools. One of them, the synthetical school, was 

 represented by Steiner, Mobius, v. Staudt, Schroter, and has as 

 its present principal leaders: Durege, Reye, Sturm, and Fiedler. 



