with a Determination of the Limit beyond which it fails. 335 
recover the four-square form, or some variety of that form, should it have 
escaped the memory. In this limited point of view, quaternions would bear 
some analogy, as respects the office performed, to the well-known contrivance of 
Napier for the solution of right-angled spherical triangles. 
But such a temporary and isolated purpose as this is not one to which qua- 
ternions are confined in Sir William Hamilton’s more comprehensive theory of 
these expressions, in which theory they are extensively employed as new and 
permanent instruments of analysis. The laws to which they are subject are, as 
we perceive from the foregoing fundamental and essential relations, altogether 
different from those which govern the operations of common algebra; and, there- 
fore, viewed as a part of common algebra, they could not be admitted. But it 
must be remembered that the symbols 2, 7, &, introduced into this theory, are 
confessedly distinct in meaning from those recognised in the older algebra; and 
are thus governed by laws peculiar to themselves. Everything in the previously 
existing algebra is left undisturbed; there is no ¢mnovation, nor, strictly speak- 
ing, any extension of its hitherto admitted principles : it is rather the addition 
to it of an algebra altogether new. Descartes, by giving a new office to certain 
marks employed in algebra, acquired additional power over symbolical geometry : 
with him the signs + and — were used to denote geometrical position. In Sir 
William Hamilton’s theory, these same signs serve to distinguish a/gebraical 
position, or order of succession ; and, in a product, mark the difference between 
taking one of the factors for a multiplier, and the other,—a distinction for which 
the ordinary algebra does not provide, but which is found to suggest new paths 
of inquiry both in pure and applied science. Carnot opposed the Cartesian 
doctrine from an imperfect view of its peculiar character, and from overlooking 
its avowed stipulations, as the author of this paper has attempted to show else- 
where ;* and it is possible that the theory of quaternions may be regarded with 
suspicion from like causes :—from an imagined discrepancy, namely, between its 
operations and the operations of the common algebra; it being altogether over- 
looked that the quantities peculiar to a quaternion are wholly distinct from those 
hitherto received into algebra, and, therefore, are not necessarily amenable to its 
* Mathematical Dissertations, Diss. I. 
VOL. XXI. DAY. 
