On the relationships between the geometric and the algebraic ideas in Duhre’s textbooks of mathematics, as reflected via Book II of Euclid’s Elements

The present article explores the relationships between the geometric and algebraic ideas presented in Anders Gabriel Duhre’s mathematics textbooks. Of particular interest is Book II of Euclid’s Elements as presented by Duhre in his textbook on geometry from 1721. We consider in detail Duhre’s two versions of Proposition II.5, dealing with straight lines cut into equal and unequal parts, as well as the two proofs of the propositions that he presents. Duhre’s formulations are slightly different from traditional geometric formulations, as he moved away from a purely geometrical context towards an algebraic one. Duhre estab- lished Proposition II.5 using algebra in Descartes’ notation as well as in the notation of Wallis and Oughtred. Duhre ́s reason for introducing algebra in Book II of Euclid’s Elements was to obtain con- venience in calculations, as well as the possibility to generalize results to different kinds of quantities