Chapter: Avoidance of Negative Solutions, Descartes' Analysis and Synthesis

In his "Géometrie" published 1637 Descartes connected geometry with algebra, inventing the mathematical formula. This was very important for the further development of analysis as we know it today. However, the term "analysis" had a different meaning for Descartes as it has today. He named by the term "analysis" the method he invented for solving geometrical1 problems. The method was: 1. Name all geometrical segments with letters 1. Distinguish between "knowns" (e.g. denoted by $a,b,c$) and an "unknown" (e.g. a denoted by $x$). 1. Find at least two independent geometrical arguments to express the unknown $x$ in terms of the knowns. 1. Build an equation with these two expressions (mathematical formula!). 1. Solve the equation for $x$.

In his "analysis", Descartes did not allow any negative solutions. He and contemporary mathematicians deprived negative numbers the right of being "real" mathematical objects because negative numbers did not correspond to lengths, areas, and volumes, which were always positive. Moreover, every "analysis" had to be completed by a "synthesis". For Descartes, the synthesis was to transform the solution of the equation back to a geometrical object. By the contemporary standards, it was not sufficient for Descartes to find the positive solutions of an equation, he also wanted to know, what kind of geometrical object (e.g. segment or curve) was created, if the unknown $x$ was calculated for different original values of the knowns. Like Galileo before him (see Renaissance and Beginnings of the Infinitesimal Methods) Descartes also failed to explain what we would call today the continuity of a function.

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  1. Spalt, Detlef: "Eine kurze Geschichte der Analysis", Springer Spektrum, 2019


  1. i.e. mathematical problems in general, since in the 17th century, only geometry was perceived as mathematics.