of Edinburgh, Session 1868-69. 
555 
mathematical part of the necessary investigations could hardly have 
been made so simple and yet comprehensive as it now is had 
quaternions not been employed. 
3. Hegel and the Metaphysics of the Fluxional Calculus. By 
W. R. Smith, Esq. Communicated by Professor Tait. 
(Abstract.) 
The object of this paper is to consider the mathematical value 
of Hegel’s discussion of the fluxional calculus. This discussion, 
as contained in several notes in the “ Logik,” professes to evolve 
the true principle of the calculus in a form free from the incon- 
sistencies of the usual processes, and to show how this principle 
may be rendered useful in the solution of problems. To clear the 
way for this new theory, Hegel engages in a sharp polemic against 
the expositions of several parts of the calculus given by Newton 
and others. 
As these strictures on Newton have, at least in part, been re- 
ceived with great satisfaction by metaphysicians, the paper seeks 
first to show that the notion on which Newton based his doctrine 
of fluxions, viz., the generation of magnitudes by continuous 
motion at definite (but not necessarily constant) velocity, really 
has a place in nature. If we remember that Newton always views 
fluxions as a method belonging to physics (kinematics), we shall 
find the whole developments of his calculus to be thoroughly con- 
sistent and simple. It is shown, however, that Hegel, while he 
professes to approve of Newton’s fundamental notions, has so com- 
pletely misunderstood the whole method as to propose to eliminate 
as inessential the conceptions of movement and velocity, and then 
to blame Newton for attempting to use the calculus to do work for 
which it is fitted just because it is based on the notion of continuous 
movement. Not only does it appear that Hegel has failed to 
appreciate Newton’s general principle, he seems also to have quite 
failed to understand what mathematicians mean by a limit. His 
strictures on the doctrine of limits are in no sense directed against 
any tenet of mathematicians, but only against certain absurdities 
which he supposes mathematicians to hold. This especially appears 
in his inability to understand what is meant by the evaluation of a 
