672 SCIENTIFIC THOUGHT. 



age quoted above shows, such an idea must have been 

 before the mind of Jacob Steiner when he wrote the 

 ' Systematische Entwickelung.' Through Euclid geo- 

 metricians had learnt to begin with the straight line of 

 definite not indefinite length, the triangle, the circle, 

 advancing to more complicated figures ; practice had 

 made geometry a science of mensuration, involving 

 number ; the convenience of practice in . astronomy, 

 geodesy, and geography had introduced the artifice of 

 referring points and figures in space to certain arbi- 

 trarily chosen data points and lines. The terms " right 

 ascension " and " declination," " altitude " and " azi- 

 muth," " latitude " and " longitude," led to the co- 

 ordinates of Descartes and to analytical geometry. In 

 this older and modern geometry, the beginnings were 

 arbitrary, and many conceptions were introduced which 

 were foreign to the object of research. It was through 

 a slow process that in quite recent times notably dur- 

 ing the nineteenth century mathematicians became 

 aware how artificial were their methods, and with how 

 many foreign elements they had encumbered the objects 

 of their study. To replace the artificial by natural con- 

 ceptions, and to open the eyes of geometricians to the 

 advantage of not confining themselves to the point (its 

 motion and distances) as the element in their space 

 construction, no one did more than Julius Pliicker of 

 Bonn. We have now not only a point - geometry, 

 but likewise a line -geometry i.e., we have a geom- 

 etry in which the line is the primary element, the 

 point being the secondary element, defined by the 

 intersection of two lines. This conception, which 



