1^94-] BAKER — GRAPHIC IMAGINARIES. I5I 



Dr. George W. Goler read a paper entitled : 

 SMALL POX AND THE PRESENT STATUS OF THE 

 VACCINATION QUESTION. 



The paper was illustrated by statistical charts, and was discussed 

 by Dr. J. L. Roseboom. 



November 12, 1894. 

 The President in the chair. Thirty-five persons present. UBR^*^^ 



The Secretary presented the following paper : NEW YORK 



GRAPHIC IMAGINARIES. ^^/^rDEN^ 



By Professor Arthur Latham Baker. 

 There was a time in the history of mathematics when a negative 

 quantity was wholly imaginary. This was when the concept of 

 number had not risen above that of counting material objects. Under 

 these circumstances, of course, a negative object, such as a negative 

 book or the negative inhabitant of a town, was purely imaginary. 



Later, when the concept of debit and credit arose, negative 

 numbers became real, to a man deeply in debt disastrously real. 

 Positive numbers represented his credits or resources and if his debts 

 outnumbered these, his profits were appallingly negative. 



For half a millennium after this, the y (-1) was also an imaginary 

 number, for the reason that when operating upon arithmetical symbols, 

 we can find no arithmetical symbol which when squared will give the 

 arithmetical symbol -i. The literature of the attempts to extend the 

 concept so that this imaginary symbol should become real is exceed- 

 ingly voluminous and it is only in comparatively recent times that the 

 attempt has been successful. Argand in the famous Argand diagram 

 first opened the way. That there must be a real interpretation, pro- 

 vided we enlarged sufiiciendy the concept, was inferable from the 

 persistency with which the symbol obtruded itself into the simplest 

 algebraic operations. 



As when we rose from the concept of only material objects to 



that of debit and credit, the imaginary negative became real, so when 



we rise from the concept of an arithmetical or algebraic symbol to a 



more comprehensive one, the imaginary j/ (-1 ) in its turn becomes real. 



I have provisionally divided symbols, as standing for mathemat- 



20, Proc. Roch. Acad, of Sc, Vol. 3, December, i8qq. 



