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PAPERS READ. 
On the imaginary of algebra. By Prof. A. Macfarlane, University 
of Texas, Austin, Texas. 
The student, if he should hereafter inquire into the assertions of different writers, 
who contend for what each of them considers as the explanation of will do well 
to substitute the indefinite article.”— De Morgan, Double Algebra, p. 94. 
With respect to the theory and use of |/—1 analysts may be divided 
into three classes: first, those who have considered it as undefined and 
uninterpreted, and consequently make use of it only in a tentative manner; 
second, those who have considered it as undefinable and uninterpretable, 
and build upon this supposed fact a special theory of reasoning; third, 
those who, viewing it as capable of definition, have sought for the defi 
nition in the ideas of geometry. 
Of the first class we have an example in the view laid down by the 
astronomer Airy (Cambridge Philosophical Transactions, vol. x, p. 327). 
“I have not the smallest confidence in any result which is essentially ob 
tained by the use of imaginary symbols. I am very glad to use them as 
conveniently indicating a conclusion which it may afterwards be possible 
to obtain by strictly logical methods; but until these logical methods 
shall have been discovered, I regard the result as requiring further dem 
onstration.” This view admits that conclusions are indicated by methods 
which are not strictly logical; that a method which is not strictly logical 
can indicate and always can indicate a conclusion is a paradox which it is 
very desirable to explain. 
Of the second class we have an example in the mathematician and logic 
ian, Boole. Instead of conforming analysis to ordinary reasoning, he 
endeavors to conform reasoning to analysis by introducing a transcend 
ental species of logic. In his Laws of Thought, p. 68, he lays down the 
following as an axiomatic principle in reasoning: The process of solu 
tion or demonstration may be conducted throughout in obedience to cer 
tain formal laws of combination of the symbols, without regard to the 
question of the interpretability of the intermediate results, provided the 
final result be interpretable. Our knowledge of the foregoing principle is 
based upon the actual occurrence of an instance, that instance being the 
imaginary of algebra. In support of this view he says : “A single example 
of reasoning in which symbols are employed in obedience to laws founded 
upon their interpretation, but without any sustained reference to that in 
terpretation, the chain of demonstration conducting us through intermedi