2 
three rectangular coordinates; while 7,7, 4 were certain ima- 
ginary units, or symbols, subject to the following laws of 
combination as regards their squares and products, 
P= ee — 1, (A) 
ij = h, jh = i, hi =}, (B) 
jinm—h, kj=—t, ho=-y, (C) 
but were entirely free from any linear relation among them- 
selves; in such a manner, that to establish an equation be- 
tween two such imaginary trinomials was to equate each of the 
three constituents, wyz, of the one to the corresponding con- 
stituent of the other; and to equate two quaternions was (in 
general) to establish rour separate and distinct equations be- 
tween real quantities. Operations on such quaternions were 
performed, as far as possible, according to the analogies of 
ordinary algebra; the distributive property of multiplication, 
and another, which may be called the associative property of 
that operation, being, for example, retained: with one impor- 
tant departure, however, from the received rules of calculation, 
arising from the abandonment of the commutative property of 
multiplication, as not in general holding good for the mixture 
of the new imaginaries; since the product ji (for example) 
has, by its definition, a different sign from 7. And several 
constructions and conclusions, especially as respected the geo- 
metry of the sphere, were drawn from these principles, of 
which some have since been printed among the Proceedings 
of the Academy for the date already referred to. 
The author has not seen cause, in his subsequent re- 
flections on the subject, to abandon any of the principles 
which have been thus briefly recapitulated; but he conceives 
that he has been enabled to present some of them in a clearer 
view, as regards their bearings on geometrical questions; and 
also to improve the algebraical method of applying them, or 
what may be called the caLcULUS OF QUATERNIONS. 
Thus he has found it useful, in many applications, to dis- 
miss the separate consideration of the three real constituents, 
