SECOND APPROXIMATION

the child’s height differs from the father’s by four timesas much as the child’s less two feet. This may sound ascircular as the first statement—the father’s height is ex-pressed in terms of the child’s, and the child’s is expressedin terms of the father’s; but the second stipulation is notnow reducible to the first. The heights are entirely deter-minate, that of the father being six feet and that of thechild, two. The mere fact that both of these magnitudes,the father’s height and the child’s height, are specifiedeach in terms of the other does not constitute a viciouscircle. The general principle, as Cournot and other mathe-matical economists have often pointed out, is simply thewell known algebraic principle of simultaneous equations.In order that the equations may determine the unknownquantities involved, there must be as many independentequations as there are unknown quantities, although anyor all of these equations may contain all the unknowns.(The equations are independent if no one of them canbe derived from another or the others.) Many an exampleof economic confusion and wrong reasoning could beavoided if this fundamental principle of mathematicswere more generally applied.

This mathematical principle of determinateness appliesin our present problem. Real examples of circular reason-ing in the theory of interest are common enough, but thedependence, above stated, of interest on the range of op-tions and the dependence of the choice among them oninterest is not a case in point, for this last determiningcondition is not derivable from the others . 5

For our present purpose we need only present thematter to the reader’s imagination by a process of trial

“That this is the case under our present hypothesis is shown fully inChapter XIII.

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