Lengths of Curves

Abstract

As was mentioned in XII.5.9, it is possible to found the theory of trigonometric functions on a study of lengths of circular arcs. Such an approach is suggested in the syllabus notes (S₁), (S₂) and (S₄) and is adopted by various high school text books. The treatment given by Mulhall and Smith-White (1₂), pp. 32–36 and (1₄), p. 22 is pretty typical and will be scrutinised at some length; see also Swokowski (1), pp. 247–248, 488–490. If this approach to measure of angles and trigonometric functions is to be acceptable and carry real conviction, the idea of lengths of curves deserves more care than is accorded to it in typical text books. Even the notion of “curve” has to be examined. A precise definition which is in general accord with the intuitive idea is not easy to formulate and justify. To cover this fully is not attempted in this book, but see the indications in Edwards (4): all that is done here is to proceed far enough to at least disperse some of the haze enveloping many high school accounts of this topic.