MATHEMATICS IN THE NINETEENTH CENTURY 491 



radicals was finally shown by the youthful Abel (1824), while Hermite 

 and Kroneker (1858) showed how they might be expressed by the 

 elliptic modular functions, and Klein (1875) by means of the icosa- 

 hedral irrationality. 



But of all problems which have come down from the past, by far 

 the most celebrated and important relates to Euclid's parallel 

 axiom. Its solution has profoundly affected our views of space, 

 and given rise to questions even deeper and more far-reaching which 

 embrace the entire foundation of geometry and our space conception. 

 Let us pass in rapid review the principal events of this great move- 

 ment. Wallis in the seventeenth, Seccheri, Lambert, and Legendre 

 in the eighteenth, are the first to make any noteworthy progress 

 before the nineteenth century. The really profound investigations 

 of Seccheri and Lambert, strangely enough, were entirely over- 

 looked by later writers and have only recently come to light. 



In the nineteenth century non-Euclidean geometry develops along 

 four directions, which roughly follow each other chronologically. 

 Let us consider them in order. 



The naive-synthetic direction. The methods employed are similar to 

 those of Euclid. His axioms are assumed with the exception of the 

 parallel axiom; the resulting geometry is what is now called hyper- 

 bolic or Lobatschewski 's geometry. Its principal properties are de- 

 duced, in particular its trigonometry, which is shown to be that of a 

 sphere with imaginary radius as Lambert had divined. As a specific 

 result of these investigations the long-debated question relating to 

 the independence of the parallel axiom was finally settled. The great 

 names in this group are Lobatschewski, Bolyai, and Gauss. The first 

 publications of Lobatschewski are his Exposition succinct des prin- 

 cipesde la geometric (1829), and the Geometrische Untersuchungen, in 

 1840. Bolyai's Appendix was published in 1832. As to the extent 

 of Gauss's investigations, we can only judge from scattered remarks 

 in private letters and his reviews of books relating to the parallel 

 axioms. His dread of the Geschrei der Sootier, that is, the followers 

 of Kant, prevented him from publishing his extensive speculations. 



The metric-differential direction. This is inaugurated by three great 

 memoirs by Riemann, Helmholtz, and Beltrami, all published in the 

 same year, 1868. 



Beltrami, making use of results of Gauss and Minding relating to 

 the applicability of two surfaces, shows that the hyperbolic geometry 

 of a plane may be interpreted on a surface of constant negative 

 curvature, the pseudosphere. By means of this discovery the purely 

 logical and hypothetical system of Lobatschewski and Bolyai takes 

 on a form as concrete and tangible as the geometry of a plane. 



The work of Riemann is as original as profound. He considers 

 space as an n-dimensional continuous numerical multiplicity, which 



