DEVELOPMENT OF MATHEMATICAL THOUGHT. 669 



nently into the foreground. The latter was done by 

 the geometric genius of Von Staudt, who succeeded in 

 giving a purely geometrical interpretation of the imagin- 

 ary or invisible elements l which algebra had introduced, 

 whilst Steiner astonished the mathematical world by the 

 fertility of the methods by which he solved the so- 

 called isoperimetrical problems i.e., problems referring 

 to largest or smallest contents contained in a given 

 perimeter or vice versd, problems for which Euler and 

 Lagrange had invented a special calculus. 2 In spite of 



1 The geometrical interpretation 

 of the imaginary elements is given 

 by Von Staudt in a sequel to his 

 ' Geometric der Lage ' (1847), en- 

 titled ' Beitrage zur Geometric der 

 Lage' (1856-60); and after hav- 

 ing been looked upon for a long 

 time as a curiosity or a " hair- 

 splitting abstraction," it has 

 latterly, through the labours of 

 Prof. Reye (' Geometric der Lage,' 

 1866-68) and Prof. Ltiroth ('Math. 

 Annalen,' vol. xiii. p. 145), become 

 more accessible, and is systematic- 

 ally introduced into many excel- 

 lent text-books published abroad. 

 The simplest exposition I am ac- 

 quainted with is to be found in 

 the later editions of Dr Fiedler's 

 German edition of Salmon's ' Conic 

 Sections' (6th Aufl., vol. i. p. 23, &c., 

 and p. 176, &c.) In 1875, before 

 the great change which has brought 

 unity and connection into many 

 isolated and fragmentary contribu- 

 tions had been recognised, Hankel 

 wrote with regard to Von Staudt's 

 work, and in comparison with that 

 of Chasles, as follows : " The work 

 of Von Staudt, classical in its 

 originality, is one of those attempts 

 to force the mauifoldness of nature 

 with its thousand threads running 

 hither and thither into an abstract 

 scheme and an artificial system : an 

 attempt such as is only possible in 



our Fatherland, a country of strict 

 scholastic method, and, we may add, 

 of scientific pedantry. The French 

 certainly do as much in the exact 

 sciences as the Germans, but they 

 take the instruments wherever 

 they find them, do not sacrifice 

 intuitive evidence to a love of 

 system nor the facility of method 

 to its purity. In the quiet town 

 of Erlangen, Von Staudt might well 

 develop for himself in seclusion 

 his scientific system, which he 

 would only now and then explain 

 at his desk to one or two pupils. 

 In Paris, in vivid intercourse with 

 colleagues and numerous pupils, 

 the elaboration of the system 

 would have been impossible " (loc. 

 cit., p. 30). 



2 See the lecture delivered by 

 Steiuer in the Berlin Academy, 

 December 1, 1836, and the two 

 memoirs on ' Maximum and Min- 

 imum' (1841), reprinted in 'Ge- 

 sainmelte Werke, 1 vol. ii. p. 75 

 qq., and 177 sqq., especially the 

 interesting Introductions to both, 

 in which he refers to his fore- 

 runner Lhuilier (1782), deploring 

 that others had needlessly forsaken 

 the simple synthetical methods 

 adopted by him. Some of 

 Steiner's expositions in these 

 matters were apparently so easy 

 that non - mathematical listeners- 



