FROM NUMBER TO QUATERNION. 65 



FROM NUMBER TO QUATERNION 



By G. Fleuri, Licencie es-sciences mathematiques and 

 Licencie es-sciences physiques, Sydney, N. S. Wales. 



[With Sixteen Diagrams.] 

 (Communicated by H. C. Russell, b.a., c.m.g., f.r.s.) 



[Bead before the Royal Society of N. S. Wales, June 6, 1894.] 



Preliminaries. — I propose to show in the present paper how 

 mathematicians impelled by that spirit of generalization which 

 predominates the whole of mathematical science may pass quite 

 naturally from the idea of number to that of quaternion. 



First of all, I will establish in a precise manner what it is that 

 distinguishes algebra from arithmetic, and that part of my paper 

 up to the introduction of quaternion may be considered as an 

 elementary exposition of the fundamental principles of algebra 

 after Grassman's* and Hankel'sf ideas. 



It is a pity that these fundamental principles of algebra should 

 be always taught in the same awkward manner as they were 

 several hundred years ago.J I am a believer in quaternions, and 



* Grassman — "Ausdehnungslehre," Stettin 1862. 



f Grassman's ideas would not yet perhaps be understood if it were not 

 for Hankel's " Vorlesungen iiber complexe Zahlen und ihre functionen/' 

 Leipzig, 1867. 



X Nay, I may say even in a more awkward manner, for if I trust an 

 historical information given in P. Kelland and Tait's " Introduction to 

 Quaternions," " Diophantus in his treatise on arithmetic boldly lays it 

 down as a definition or first principle of his science that minus into minus 

 makes plus" and a professor of mathematics to whom I was speaking of 

 the impossibility of the demonstration of that fact by old methods, told 

 me that " having first shown the students that minus into plus makes 

 minus he concludes that minus changed the signs and therefore changed 

 minus into plus," a reasoning equivalent to the following : — *' a blue 

 liquid into a yellow liquid gives a green liquid and therefore changes 

 the color, so that aVuj liquid into a blue liquid must give a color differ- 

 ent from blue. 



E— June 6, 1E94. 



