THE INDUCTIVE PROCESS. 
125 
as one of the most firmly established is the Law of Causation, the 
modern method — for the systemization of which we are so much 
indebted to Mr. Mill — connecting small and weak inductions with 
this great and firm one, possesses a vast superiority over the ancient 
by so greatly increasing the warrant for believing the results of 
such otherwise precarious reasoning. 
It must, however, be remembered that our inductive conclusions 
being thus of the nature of probabilities, and as such specifically 
different from actual certainties, are not capable of being used, like 
the first principles of mathematics, as a basis for an immense 
superstructure of deductive reasoning, except to such an extent 
only as the results may be verified by further observation or 
experiment. 
Are then the mathematical sciences apodictic ? 
Perfectly so as long as they remain pure. And the lecturer is 
inclined to believe that they derive this peculiar property from the 
formal certainty of the induction by complete enumeration. They 
are all strict deductions from a very few first principles exceedingly 
simple, which although suggested to the mind by experience of 
concrete instances, are no longer the mere mirrored reflexion of 
those fallible cases, but have become purely subjective abstractions, 
superior to and independent of the concrete ; and being thus wholly 
contained in the mind, and of a limited character, have been made 
the subject of a complete examination, which cannot be done with 
the unlimited object world. They retain their apodictic character 
only so long as they remain strictly subjective, and are often liable 
to defeat when applied to the concrete. We can postulate as a 
subjective possibility the existence of two parallel lines, but can 
never satisfy ourselves by measurement that any two actual lines 
are rigorously such. So too it is impossible in the abstract, but 
that when equals are added to equals the wholes shall be equal ; 
but if we test this axiom as regards space by adding together two 
pints of water on one side of the equation and a pint each of water 
and sulphuric acid on the other, it will be found that its applica- 
tion is defeated, although it is from concrete objects like a pint of 
liquid that the abstract pint of space was first conceived. The 
pure mathematics are purely subjective, which is the real ground 
of Sir William Hamilton's objection to the influence of an undue 
preponderance of mathematical study in the formation of character. 
The lecturer concluded by applying his views to show the 
