CHARLES B. WARRING. 113 
And besides this, some mathematical truths otherwise 
not easily demonstrated are obtained by theirhelp. The 
following example, taken in substance from the ninth 
edition of the Encyclopedia Britannica, will suffice for 
an illustration : 
It is required to demonstrate that the product of the 
sum of two squares is itself the sum of two other 
squares. In other words, it is required to prove that 
(a’-|-6*)--(e'-+-d@’) equals the sum of two squares. 
(1). Since (a+tyv —1)(~v—yvV —1)=2'+y’, we may as- 
sume Sans poet 
(2). a@+0°=(a--by —1)(a—by —1). 
(3). ¢+d’=(ce+dy —1)(e—dV —1). 
(2) (3)=(a+bv —1)(e—dv —1)(a—by -1)(e+-dv = 1) 
Or, (ac—adv —1-+-bev —1-++-dd) (ac--advV —1—bev —1 
+d). 
Or, (ac+bd+be—adv —1) (ac+bd—bc—adv —1). 
If we put ac+bd=A, 
and b¢c—ad=B, we have 
(2) X(3)=(A+B V—-1) (A—By-—1) 
or, (@°+0*) (c’+d*)= A’+B’. Q. E. D. 
What then can be done to reconcile our minds to the 
apparently illogical result that, from unreal, impossible 
premises, come real and true conclusions ? 
To aid in doing this, it has been attempted to represent 
the impossible (or ‘‘ imaginary ’’) quantities graphically. 
It has been said that Multiplication, in its broad sense, 
is not merely, as in Arithmetic, a process of repeating 
the Multiplicand, but, in algebra, it includes another 
idea, that of direction, and multiplying by —1 reverses 
the direction of the multiplicand, if this be represented 
by aline. Starting from «a certain point called the ori- 
gin, we agree that all toward the right shall be called 
positive, and all toward the left negative. Now, if we 
have one or more + units, ¢.¢., to the right, and multiply 
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