The fusion of plane and solid geometry in the teaching of geometry: Textbooks, aims, discussions

The idea of the fusion of plane and solid geometry originated from projective and descriptive geometry, which worked with projections in space and sections. Different authors of textbooks (starting from Bretschneider in 1844 to Méray in 1874/1903; de Paolis in 1884; Lazzeri & Bassani in 1891, also translated into German by Treutlein in 1911) adopted this idea, mixing plane and solid considerations. For instance, the chapter on the properties of incidence also referred to the mutual position of a plane and a straight line, while homothety was defined in space and then on the plane. Pupils were supposed to have a better intuition of spatial relations when passing from space to plane, and to reason by analogy. Moreover, proofs could be presented of plane theorems using projections in space of simple known configurations. In the textbook of Lazzeri and Bassani we can see that one of the aims of the authors is to prove plane theorems with the help of considerations in space that allow to avoid part of the congruence axioms and the theory of proportions. This is not a novelty within history of mathematics, the development of conic sections is linked to this point, and Monge, too, used it in 1799.