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rection of Forward, but that of Negative Unity to the same 
magnitude measured backward; and if we extend to this po- 
sitive unity and to lines having direction in space the received 
definitions of multiplication, that ‘‘ Positive Unity is to Mul- 
tiplier as Multiplicand is to Product,” and that ‘‘ the product 
of two equal factors is the square of either ;” we may then 
consistently and naturally be led to assert the same results as 
those already enunciated from the theory of quaternions re- 
specting the product of two vectors, in the two principal 
cases, first, where those two vectors are rectangular, and se- 
cond, where they are coincident with each other. And thus 
may we justify, or at least interpret and explain, the funda- 
mental definitions (A) (B) (C) of this theory, by regarding 
the symbols #jk as denoting three vector-units having three 
rectangular directions in space, 
But farther, we derive from this view of the whole subject 
an illustration (if not a confirmation) of the remarkable con- 
clusion that the so-called real and positive unit + 1 is not (in 
this theory) to be confounded with any vector unit whatever, 
but is to be regarded as of a kind essentially distinct from 
every vector. For this positive unit + 1 is in the direction 
above called Forward, and denoted by A. Now if this could 
coincide with a direction X in tridimensional space, then, 
whatever this latter direction might be supposed to be, we 
_ eould always, by the general formula A: X:: Y: Z (where X 
is arbitrary), deduce the inadmissible proportion X: X:: Y:: Z, 
in which the two directions in one ratio are identical, but 
those in the other are rectangular to each other. If then we 
resolve to retain the assumption of the existence of a fourth 
proportional A to three rectangular directions in space, as 
subject to be reasoned on at all in the way already described, 
and as determined in direction by its contrast to its own oppo- 
site B (corresponding to an opposite order of rotation in the 
system X YZ), we must think of these two opposite directions 
A and B as merely laid down upon a scale, but must abstain 
