DEVELOPMENT OF MATHEMATICAL THOUGHT. 677 



forms, and to have foreseen its importance and corre- 

 sponding significance when applied to a great variety of 

 scientific problems, notably to the projective processes 

 in geometry. These were known to them mainly 

 through the classical treatises of Poncelet and Chasles, 

 the leading ideas of which had been introduced to 

 British students by the labours of the Dublin school. 1 

 The investigations referred to mark the junction of 

 two important lines of mathematical research, which 

 had been carried on independently in earlier times, or 

 only united for special purposes or for the solution 

 of special problems. The history of the progress of 

 geometry during the nineteenth century has already 

 shown us the use and interest which belong to two 

 different aspects of the common object, of which the 

 one relies mainly on processes of measurement, including 

 number, the other mainly on processes of description, in- 



1 The history of the doctrine of 

 invariants has been written by Dr 

 Franz Meyer, and is published in the 

 first volume of the ' Jahresbericht 

 der Deutschen Mathematiker Ver- 

 einigung ' (p. 79 sqq.) The fact that 

 this formed the first of the several 

 Reports which the German Mathe- 

 matical Society has undertaken to 

 publish, testifies to the great im- 

 portance which belongs to this 

 doctrine in the history of recent 

 mathematics. A concise summary 

 with copious references is given by 

 the same author in the first volume 

 of the ' Encyklopadie der Math. 

 "\Vi.ssenschaften,' p. 320 sqq. How 

 necessary the form and perfection 

 of algebraic operations was for the 

 development of the geometrical 

 conceptions which are laid down, 

 e.g., in the works of Plucker, 

 can be seen in the work of 

 Otto Hesse, who introduced ele- 



gance and conciseness into many 

 of the expositions which, for want 

 of this formal development, ap- 

 pear cumbrous in the writings of 

 Plucker. " The analytical form in 

 which Pliicker's Researches present 

 themselves is frequently wanting 

 in that elegant form to which we 

 have become accustomed, specially 

 through Hesse. Pliicker's calcula- 

 tions frequently bear the stamp of 

 mere aids for representing geo- 

 metrical relations. That algebraical 

 connections possess an interest in 

 themselves, and require an ade- 

 quate representation, was realised 

 only by a generation which habitu- 

 ally employed methods that had 

 been largely devised by Plucker 

 himself " (A. Clebsch, ' Zum Ge- 

 dachtniss an Julius Plucker, ' 1872, 

 p. 8. See also Gustav Bauer, 

 ' Gedachtnissrede auf Otto Hesse,' 

 Miinchen, 1882). 



