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 does to the figures made up “by putting together many short straight lines.”  Correspondingly, the Proper Name may be called the Limit of the class-name; since its attributes are infinite, whereas any name whose attributes are less than infinite stands for a possible class.  In short, for logical purposes, a Limit may be defined as any extreme case to which actual examples may approach without ever reaching it.  And in this sense ‘Method of Limits’ might be used as a term including the Method of Abstractions; though it would be better to speak of them generically as ‘Methods of Approximation.’

We may also notice the Assumptions (as they may be called) that are sometimes employed to facilitate an investigation, because some definite ground must be taken and nothing better can be thought of:  as in estimating national wealth, that furniture is half the value of the houses.

It is easy to conceive of an objector urging that such devices as the above are merely ways of avoiding the actual problems, and that they display more cunning than skill.  But science, like good sense, puts up with the best that can be had; and, like prudence, does not reject the half-loaf.  The position, that a conceivable case that can be dealt with may, under certain conditions, be substituted for one that is unworkable, is a touchstone of intelligence.  To stand out for ideals that are known to be impossible, is only an excuse for doing nothing at all.