DEVELOPMENT OF MATHEMATICAL THOUGHT. 685 



modelling of the text-books and school-books of algebra 

 and geometry in this country and in Germany, belongs 

 undeniably to Dr Salmon of Dublin. 1 The conception of 

 a form be this geometrical or algebraic suggests the 

 investigation of the change, the recurrence of forms. 

 How do forms under the process of geometrical or 

 algebraical manipulation alter or preserve their various 

 properties ? The processes of projection practised by 

 Monge, Poncelet, and Chasles in France had already 

 led to a distinction between descriptive and metrical 

 properties of geometrical figures. A corresponding ex- 

 amination of algebraical forms, which are all capable of 

 geometrical representation or interpretation, would lead 

 to the extensive and fundamental doctrine of the in- 

 variants of these forms i.e., of such arrangements of 

 the elements as remain absolutely or proportionally un- 

 altered during the processes of change and combination. 

 Notably instead of the geometrical process of projection 

 by central perspective we may employ in our algebraic 

 formulae a corresponding process, that which is known as 

 linear substitution. And at the time when it was 

 recognised that geometrical transformation had its 



1 Of Dr Salmon, whose ' Les- 

 sons introductory to the Modern 

 Higher Algebra ' appeared in 1859 

 (4th ed., 1855 ; 1st German ed. by 

 Fiedler, 1863), Meyer says : " Re- 

 cognising how the special results 

 in this domain gradually acquired 

 a considerable bulk, we must the 

 more gratefully acknowledge the 

 work of Salmon who had already, 

 in the direction of algebra as well as 

 of geometry, furnished valuable con- 

 tributions of his own in under- 

 taking the labour of collecting the 



widely-scattered material in a con- 

 cise monograph. For the promulga- 

 tion in Germany we have to thank 

 Fiedler both for his edition of 

 Salmon, and for having already 

 given an independent introduction 

 to the subject, in which especially he 

 made Cayley's applications to pro- 

 jective geometry generally access- 

 ible. About the same time (1862) 

 there appeared likewise an edition 

 by Brioschi, which gained many ad- 

 herents for the theory of Invariants 

 in Italy." 



