354 
Hamilton invented a theory of algebraic 
couplets, but the success of the invention is 
doubtful. In his presidential address be- 
fore the British Association the late Pro- 
fessor Cayley said that he could not ap- 
preciate the manner in which Hamilton 
connected algebra with the notion of time, 
and still less could he appreciate the man- 
ner in which he connected his algebraical 
couplet with the notion of time. Whether 
Hamilton has effected the explanation or 
not, it appears to be logically possible, for 
a complex quantity can be represented by 
two segments of one and the same straight 
line. 
But, be that as it may, Hamilton was led 
from algebraic couplets to algebraic triplets 
and to the problem of adapting triplets to 
the representation of lines in space. His 
guiding idea was to extend to space the 
mode of multiplication of lines in a plane 
already discovered by Argand, Warren 
and others; and it was here that he stepped 
from the time basis to the space basis—that 
is, passed from a unidimensional to a tri- 
dimensional subject, the latter including 
the former as a special case. To his sur- 
prise, he found that the multiplication of 
two lines in space, either one being ex- 
pressed in terms of three elements, led to a 
product composed not of three, but of four 
elements; and this result he deemed so 
novel and characteristic that he selected it 
to give a name to the new method—‘ Qua- 
ternions.’ As finally developed, the method 
rests on a geouetrical basis ; nevertheless it 
is the logical generalization of ordinary 
algebra,.for the distinctive theorems of 
algebra, such as the exponential, binomial 
and multinomial theorems, have their gen- 
eralized counterparts in quaternions. Since 
the time of Gauss, mathematicians have 
considered double or plane algebra to be 
the logical generalization of ordinary al- 
gebra ; now quaternions bears to plane al- 
gebra the same logical relation which plane 
SCIENCE. 
[N. 8. Von. X. No. 246. 
algebra bears to ordinary algebra. It is all 
algebra in the sense of being the analysis of 
quantity and the relations of quantities. 
Any one who admits De Moivre’s theorem 
into algebra is logically bound to admit 
quaternions as the highest form of algebra. 
It is a common belief that quaternions has 
only a remote connection with algebra ; 
that it is only one of several systems of non- 
commutative algebra, and that the mathe- 
matician can get on very well without it. 
But if the above is the true logical relation, 
then it must be the duty of every analyst to 
master its principles. It may be remarked 
here that the logical relation of quaternions 
to plane algebra is obscured by the preva- 
lent but erroneous idea that the complex 
quantities of the form x + wy represent vec- 
tors. They really represent, in their planar 
meaning, coaxial quaternions ; that is, x is 
a scalar and the axis of y is the common 
perpendicular to the plane. Let, as usual,, 
w+ ix + jy + kz denote a quaternion ; the 
complex quantity is identical not with 
wtia« or ix+ jy, but with w+kz. The 
fallacy in question almost baffled Hamilton 
in his attempts at generalization, as may be 
seen from the account which he gives of 
the discovery in the Philosophical Magazine 
for 1844. 
_We shall obtain additional insight into 
the nature of the fundamental laws of al- 
gebra by considering the part which they 
played in the discovery of the quaternion 
generalization. In the endeavor to adapt 
the general conception of a triplet to the 
multiplication of lines in space Hamilton 
started out with the principles of commu- 
tation, distribution and reduction ; but in 
order that the theorem about the moduli 
might remain true he soon felt obliged, not 
indeed to abandon the principle of commu- 
tation entirely, but to modify it so as to 
preserve the order of the factors while leav- 
ing the order of combination of the factors 
commutable. This principle, which had 
