Pieri’s Point and Sphere Memoir

Abstract

This chapter contains an English translation of Pieri’s 1908a memoir, Elementary Geometry Based on the Notions of Point and Sphere.¹ The work had two main goals. First, it presented elementary Euclidean geometry as a hypothetical-deductive system, and showed that all its notions and postulates can be defined and formulated in terms of the notion point and the relation that holds between points a,b,c just when a,b are equidistant from c. As noted in section 5.2, this result gave rise, over decades, to a stream of related research that still continues. The paper’s title reflects Pieri’s extensive use of elementary set theory in developing geometry from his postulates: he defined the sphere through b centered at c as the set of all points a such that a and b are equidistant from c. Pieri’s second aim was to foster more extensive use of properties of spheres in presenting elementary geometry, even in school courses. In this regard, he seems to have had less impact, even though this memoir presents many useful examples. A third aim, which Pieri had already pursued for a decade, was to promote the use of transformations in elementary geometry. Pieri introduced various geometric transformations early through definitions, and employed them extensively throughout the paper, following paths already explored in his 1900a Point and Motion memoir. Finally, Pieri followed the strategy of fusionism in developing plane and solid geometry together.²

Pieri 1908a is the same as the version in Pieri’s collected works on foundations of mathematics (1980, 455–560) except for pagination and the reference to Guido Castelnuovo and Corrado Segre on the title page. The memoir has also been translated into Polish (Pieri 1914).

Fusionism is discussed in section 2.5 of the present book. Pieri subtly reinforced its practice by altering his initial quotation from Staudt 1847.