CHARLES B. WARRING. 115 
possible, or imaginary quantities that it gives a line which 
is neither larger, nor smaller, nor equal to certain other 
lines inthesame diagram. It is true that such a relation, 
or rather such a lack of relation, can be predicated of 
é. g., color and weight, or music and painting, but we do 
not attempt any graphical representation which brings 
both into one diagram. 
Whatever advantages there may be in this mode of 
representing imaginary quantities, it explains nothing, 
_for it is itself founded on the postulate thatv—1xv—1= 
the actual quantity —1, and itis in this that the real 
difficulty lies. How is it that out of quantities that do 
not exist, real results arise ? 
Thus, e. g., if I multiply 54v —1 by 5—4v—1, I 
get 41; or, omitting the real number, I multiply say 
—,/— 16 by +/—4, I get the real and positive number 8. 
I take something which does not exist, and multiply it 
by a number that does not exist, and I get something 
which does exist. 
I think the explanation of these results lies in the 
principle that one false supposition may be corrected 
by another. Asin grammar, we say two negatives make 
an affirmative, so in mathematics two suppositions, both 
false, may give a truth. One falsehood in our reasoning 
will vitiate our conclusion, but two may neutralize 
each other. If they do, then our conclusion will be 
true. If, for instance, I say that I stepped this even- 
ing from my office across the Atlantic into London,. 
the conclusion would be that I am now in London, a 
false one. But if ladd that I stepped from there into 
this, our Institute, then the logical conclusion would be 
that I came this evening from my office to this place, and 
this is the truth. I may put into my reasoning as many 
of these misstatements as I please ; and if, in mathemati- 
cal phrase, their sum is zero, my conclusion will be true. 
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