656 SCIENTIFIC THOUGHT. 



arithmetic based upon two units instead of one — i.e., the 

 arithmetic of couples or complex quantities — could be 

 completely and consistently represented by choosing as 

 axes whereon the separate units were counted, the two 

 perpendicular axes of Cartesian geometry. An attempt 

 to extend this geometrical representation into space led 

 Hamilton to the invention of his method, Gauss having 

 very early satisfied himself that within the limits of 

 ordinary algebra no further extension was necessary or 

 possible. 



The examination into fundamental principles was not 



limited in the mind of Gauss to those of algebra : he 



early applied himself likewise to those of geometry and 



of dynamics. The great French mathematicians, such 



21. as Legendre and Lagrange, were also occupied with such 



Foundations ® _ o o ' r 



of geometry, speculations. They have been carried on all through 

 the century, but have only towards the end of the 

 period been brought into connection and shown to be 

 of importance for the general progress of mathematics. 

 The secluded, and for a long time unappreciated, labours 

 of isolated but highly original thinkers have accordingly 



considered as merely special in- ' Grassmann's works is being pub- 

 stances. This has now been abund- ' lished by Teubner. Those who are 



antly proved through the writings 

 of mathematicians in all countries, 

 among whom I will only mention 

 Hankel and Dr V. Schlegel in Ger- 

 many, Clifford, Prof. Henrici, and 

 latterly Mr Whitehead in England, 

 Prof. Peano in Italy, and M. Burali 

 Forti in France. See on the whole 

 subject, on the fate of Grassmann 

 and of his great work, V. Schlegel, 

 ' Die Grassmann'sche Ausdehnungs- 

 lehre,' Leipzig, 1896 ; also, by the 

 same author, a short biography of 

 Grassmann (Leipzig, Brockhaus, 

 1878). A complete edition of 



interested in seeing how the notions 

 underlying the directional calculus 

 are gradually becoming clarified, and 

 the terminology and notation settled, 

 may read with profit the controversy 

 carried on in the pages of ' Nature,' 

 vols, xlvii. and xlviii., between Pi'of. 

 Macfarlane, Willard Gibbs, Mr 0. 

 Heaviside, Mr A. M'Aulay, and Dr 

 Knott ; also Dr Larmor's review of 

 Hay ward's ' Algebra of Coplanar 

 Vectors' (vol. xlvii. p. 266), and 

 Sir R. S. Ball's reference to the 

 ' Ausdehnungslehre ' of Grassmana 

 (vol. xlviii. p. 391, 1893). 



