Leonard M. Blumenthal

A Budget of Curiosa Metrica

Nearly twenty years ago, this MONTHLY published an article by the writer devoted to those remarkable mathematical paradoxes due to Sierpinski, Mazurkiewicz, Hausdorff, Banach, and Tarski [4]. No such startling results are contained in the present paper, the title of which was obviously suggested by De Morgans famous Budget, but perhaps the reader with sufficient good will might agree that at least some of its material explains (if it does not excuse) the use of the term curiosa.

New Metric Postulates for Elliptic n-Space

Publisher Summary A distance space is formed from an abstract set S by mapping the set of all ordered pairs of elements of S into a second set, which can be a subset of S. Distance spaces are particularized by specifying the distance sets and by postulating properties of the mapping. This chapter deals with an investigation of weak four-point properties that arise in the metric study of elliptic spaces. The metric features of elliptic space that make inapplicable the methods used in the metric characterizations of other classical spaces include distinction between congruence and superposability, distinction between contained in and congruently contained in, dependence not a congruence invariant, non-linearity of the equidistant locus, and cardinality of the maximal equilateral set.