ABSTRACT
This chapter discusses two questions. First, can projective and descriptive Geometry be established without any metrical presuppositions, or even without implying metrical properties? Secondly, can metrical Geometry be deduced from either of the others, or, if not, what unavoidable novelties does it introduce? Although the usual so-called projective theory of distance, both in descriptive and in projective space, is purely technical, yet such spaces do necessarily possess metrical properties, which can be defined and deduced without new indefinables or indemonstrables. But metrical Geometry, as an independent subject, requires the new idea of the magnitude of divisibility of a series, which is indefinable, and does not belong, properly speaking, to pure mathematics. Projective and descriptive Geometry are both independent of all metrical assumptions, and allow the development of metrical properties out of themselves.
