18 G. H. KNIBBS. 



[1796 - 1863]. 1 Steiner's 'systematic development of the depend- 

 ence of geometrical forms upon one another'; Chasles' [1793 - 

 1880] 'enumerative geometry,' afterwards extended by Hermann 

 Schubert to w-dimensional space; von Staudt's [1798-1867], 

 'geometry of position, 'interpreted by Reye, for it was too condensed 

 for the average reader ; Luigi Cremona's ' introduction to a geo- 

 metrical theory of plane curves;' Karl Culmann's graphical statics . 

 Lobatchewsky's and Wolfgang Bolyais' imaginary and absolute 

 geometries; Johann Bolyais' science absolute of space; Riem'ann's 

 thesis on w-ply extended magnitude ; Beltrami's ' Essay on the 

 interpretation of non-euclidean geometry,' and the investigations 

 of Klein, Cayley and others, reveal with cumulative significance 

 the splendour and reach of synthetic geometry. 



The foundation of modern analytic geometry was laid by Pliicker 

 [1801 - 1868] and exposed in his ' New geometry of space founded 

 on the consideration of the straight line as space-element.' Hesse 

 [1811 — 1874] applied the ' determinant ' advances made in algebra 

 to the analytic study of curves of the third order. The results of 

 these researches were carried forward by Cayley, Salmon, and 

 Sylvester. Clebsch [1833 - 1872] shewed that 'abelian functions' 

 and geometry could be mutually helpful. The theory of surfaces 

 was developed by Serret [1819 - 1855], Kummer, Hamilton, 

 Gauss, Lie and others. 



Modern times have witnessed the birth of new and more 

 generalised algebras. Among the great labourers in that sphere 

 may be mentioned : — De Morgan [1806 - 1871] — whose ' double 

 algebra ' and ' calculus of functions,' will be remembered by all 

 interested in higher mathematics : — W. R. Hamilton [1805 - 1865] 

 whose renown depends no less perhaps upon his prediction of 

 'conical refraction' than upon his greater discovery of quaternions, 

 popularised by P. G. Tait: — Hermann Grassmann [1809- 1877] 

 whose splendid work 'Lineale Ausdehnungslehre' contains in 

 addition to a vector algebra, a geometrical algebra of wide applica- 



l It may be mentioned that Steiner learned to write only when four- 

 teen years of age. 



