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To better understand the connection between the stages of zetetics and exegetics, which roughly correspond to the ancient stages of analysis and synthesis, consider the problem of identifying two mean proportionals. Moreover, there were questions about the connection Viète forged between algebra and geometry. For Descartes in particular, there were questions of whether there was a deeper, more fundamental connection that could be forged between the solutions of algebraic problems that were expressed in terms of equations and the solutions of geometrical problems that required the construction of curves. Namely, just as some problems of arithmetic "can be imagined but not be solved," so too in geometry, there is a class of problems that require curves that are "certainly only imaginary," i.e., curves generated by "diverse motions," and thus that are not geometrical in a proper sense. In other words, there was an assumed equivalence in Viète’s program between solving an algebraic problem that required identifying the roots of specified cubic equations and solving a geometrical problem that required the construction of a curve. And it is here that the neusis postulate supplies the guarantee that such a solution can be found: By assuming the neusis problem solved, we can construct the curve that satisfies the two cubic equations above (i.e., we can construct the roots of the equations) and thereby construct the sought after mean proportionals.

Should we, as Viète urged, accept the neusis postulate as "not difficult" and thus as a foundational construction principle for geometry? That is to say, in analysis we assume what is sought as if it has been achieved, and look for the thing from which it follows, and again what comes before that, until by regressing in this way we come upon some one of the things that are already known, or that occupy the rank of a first principle. They come up with versatile instruments for analysis that can be used in diverse applications. Complete instructions on creating an instruments for analysis log file (also called a trace file) are beyond the scope of this guide. So also I hope to show instruments for analysis continuous quantities that some problems can be solved by straight lines and circles alone; others only by other curved lines, which, however, result from a single motion and can therefore be drawn with new types of compasses, which are no less exact and geometrical, I think, than the common ones used to draw circles; and finally others that can be solved by curved lines generated by diverse motions not subordinated to one another, which curves are certainly only imaginary such as the rather well-known quadratrix.

We notice in Descartes’ remarks concerning geometry in particular that the "entirely new science" he proposes will provide an exhaustive classification for problem-solving, where each of his three classes is determined by the curves needed for solution. And as with his remarks concerning the construction of geometrical curves, there were ambiguities in his discussion, which motivated varying interpretations of the method and its application to geometrical problems. In this respect, Descartes is moving from Pappus’s descriptive classification to a normative one that separates geometrical curves from non-geometrical curves, and thereby distinguishes problems that have a legitimate geometrical solution from those that do not. To make these connections clear, the brief narrative below emphasizes the proposals Descartes made during the 1618-1629 period concerning (1) the criteria for geometrical curves and legitimately geometrical constructions, and (2) the relationship between algebra and geometry. Viète’s program nicely illustrates the merging of algebra with geometrical problem-solving in early modern mathematics, and moreover, nicely illustrates an influential way of interpreting Pappus’s claims in the Collection regarding how a mathematician should apply the methods of analysis and synthesis in geometrical problem-solving. Then, the mathematician reverses the steps, and through synthesis, sets out "in natural order" the deduction leading from what is known to what is sought after.

The mathematician begins by assuming what is sought after as if it has been achieved until, through analysis, she reaches something that is already known. In the zetetic (analytic) stage of Viète’s analysis, we follow Pappus’s directive to treat "what is sought as if it has been achieved" precisely by naming the unknowns by variables. In this zetetic stage of analysis, the geometrical problem is transformed into the algebraic problem of solving a standard-form cubic equation (i.e., a cubic equation that does not include a quadratic term). In the first two cases, as Descartes treats the angular section and mean proportional problems, the compasses on which he relies are used to generate a curve that will solve the problem at hand. A similar approach is taken by Descartes when he treats the problem of constructing mean proportionals, where in this case, he appeals to his famous mesolabe compass, an instrument that is used in Book Three of the La Géométrie to solve the same problem. Basing thereupon the benefits of a multilateral ABS approach can be compared to the bilateral ABS approach. These miniature devices that are quick, precise, and easy to use offer unique benefits that cannot be matched by more traditional larger-scale equipment.