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On real functions and parametric curves (Part I)
In a series of papers which begins with the present one the author wishes to develop a theory of real functions and parametric curves and to complete thereby the results obtained in his recent research [1] on the bend of curves. Though study of real functions will occupy a large por tion of the theory and constitute by itself a unified account, the essential objective of that portion will be meant to clear the ground before we can set about full discussion of curve geometry. On the other hand we have a mind to make of our theory of curves a guiding model for possible future investigations of ours on the geometry of parametric surfaces; while we fear at the same time that our intention may not be so successfully re alized as curve length theory has served for the last several decades as a useful guiding model for the construction of surface area theory. At any rate, it would not be totally unreasonable, in view of the striking analogy between length theory and area theory, to expect the possibility of defin ing for paraetric surfaces a geometric quantity which corresponds in a way to the bend of curves and whose properties are more or less similar to those of bend. The present part is of preliminary character and deals with interval functions of certain general types and with Lebesgue-Stieltjes integration with respect to measures induced by such functions, our principal result being embodied in the theorem of • ̃ 32. The Saps treatise [7] will be quoted very frequently in the course of our study and we shall simply refer to it as Saks for brevity. Other re ferences will often be made in such forms as [1]• ̃35, for example. As regards terminology and notation we shall make it a rule to follow Saks and Iseki [1], and whenever we may deviate from the usages in these we shall each time state explicitly what we mean.
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Further measure-theoretic results in curve geometry
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Further results in Lebesgue geometry of curves
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Algebraic formulation of propositional calculi with general detachment rule
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Axiom systems of Aristotle traditional logic