Science for the multitude: Jules Dalsème's (1845-1904) “natural geometry” for the education of all

The enlargement of scientific literacy in the 19th-20th century Europe involved both new components of educational systems (primary schools and technical/vocational schools), and vulgarization of the sciences. We reconstruct the design of an intuitive, natural geometry curriculum by Jules Dalsème (1845-1904), professor of Mathematics at the Paris educational institution for prospective primary school teachers École Normale de la Seine since 1872. Dalsème's educational outlook is rooted in his engineering education (École Polytechnique) and reflects the interplay between Euclidean geometry and practical geometry for the arts and crafts. He took advantage of the concrete and reasoned geometry put forward by the engineer Édouard Lagout (1820-1884) to train workers (but also for general primary education), inspired by the activities in the construction yard. Geometry was a path for the initiation to science, as well as useful knowledge for the arts and crafts: old and new purposes of elementary schools emerge in Dalsème's textbooks for elementary school as crucial aspects for developing a mathematics for the multitude in the late modern age in Europe. His contribution is part of a more general trend going beyond numeracy and measure recipes in school mathematics and aimed to educate “reasoning citizens” and to help the nation social and economic progress.