 Book One commences with the geometrical interpretation of algebraic operations, which, we saw above, Descartes had already explored in the early period of his mathematical research. However, what we are presented in 1637 is, as Guicciardini aptly describes, a “gigantic innovation” both over Descartes’ previous work and the work of his contemporaries (Guicciardini 2009, 38). On the one hand, Descartes offers a geometrical interpretation of root extraction and thus treats five arithmetical operations (as opposed to the four operations of addition, subtraction, multiplication, and division that were treated in his early work). On the other hand, and more significantly, his treatment relies on an interpretation of arithmetical operations according to which these operations are taken to be closed operations on line segments. Traditionally, for instance, the product of two segments \(a \times b\) was interpreted as a rectangle, but for Descartes, the product is interpreted as a segment. This allows Descartes to translate geometrical problems into equations (that include products such as \(a \times b)\) and treat each term of the equation as similar in kind. Finally, Descartes uses a new exponential notation as he sets forth equations of multiple terms in Book One, and this notation, which replaces the traditional cossic notation of early modern algebra, allows Descartes to tighten the connection between algebra and geometry, and more specifically, between the algebraic representation of curves through equations with the geometrical classification and geometrical solution of stated problems (as we will see more clearly below in  section 3.2).