c methods, pains should be taken to make it clear what the
abstractions are, what are the consequent limitations upon the argument
and its conclusions, and what corrections and allowances are necessary
in order to turn the conclusions into an adequate account of the
concrete facts. The greater the number, variety, and subtlety of the
properties possessed by any object (such as human nature), the greater
are the qualifications required in the conclusions of abstract
reasoning, before they can hold true of such an object in practical
affairs.

Closely allied to this method of Abstraction is the Mathematical Method
of Limits. In his _History of Scientific Ideas_ (B. II. c. 12), Whewell
says: "The _Idea of a Limit_ supplies a new mode of establishing
mathematical truths. Thus with regard to the length of any portion of a
curve, a problem which we have just mentioned; a curve is not made up of
straight lines, and therefore we cannot by means of any of the
doctrines of elementary geometry measure the length of any curve. But we
may make up a figure nearly resembling any curve by putting together
many short straight lines, just as a polygonal building of very many
sides may nearly resemble a circular room. And in order to approach
nearer and nearer to a curve, we may make the sides more and more small,
more and more numerous. We may then possibly find some mode of
measurement, some relation of these small lines to other lines, which is
not disturbed by the multiplication of the sides, however far it be
carried. And thus we may do what is equivalent to measuring the curve
itself; for by multiplying the sides we may approach more and more
closely to the curve till no appreciable difference remains. The curve
line is the _Limit_ of the polygon; and in this process we proceed on
the _Axiom_ that 'What is true up to the Limit is true at the Limit.'"

What Whewell calls the Axiom here, others might call an Hypothesis; but
perhaps it is properly a Postulate. And it is just the obverse of the
Postulate implied in the Method of Abstractions, namely, that 'What is
true of the Abstraction is true of concrete cases the more nearly they
approach the Abstraction.' What is true of the 'Economic Man' is truer
of a broker than of a farmer, of a farmer than of a labourer, of a
labourer than of the artist of romance. Hence the Abstraction may be
called a Limit or limiting case, in the sense that it stands to concrete
individuals, as a curve doe