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April 29, 2016
These works consist almost entirely of page after page of systematic algebraic calculations and manipulations. The main emphasis is on the algebraic-arithmetic aspects of solving equations, but Viete sees this as going hand in hand with their geometrical equivalents:

"[The complete art of analysis finds solutions] both with numbers, if the problem to be solved concerns a term that is to be extracted numerically, and with lengths, surfaces or bodies, if it is a matter of exhibiting a magnitude itself. In the latter case the analyst turns geometer by executing a true construction after having worked out a solution that is analogous to the true. In the former he becomes an arithmetician, solving numerically whatever powers, either pure or affected, are exhibited." (29) "Nearly everything that is useful to the geometer for easy solution is also useful to the arithmetician, and vice versa." (236)

Thus he spells out not only how to solve equations numerically but also how anything corresponding to first or second degree algebra can be done geometrically with ruler and compass (371-387), and anything corresponding to third or fourth degree algebra by neusis (388-417).

A remarkable result of Viete's systematic analysis of equations of degree three and four shows that all such problems can be reduced to the classical problems of cube duplication or angle trisection:

"All cubic and biquadratic equations, however affected, that are not otherwise solvable can be explained in terms of two problems---one the discovery of two means between given [extremes], the other the sectioning of a given angle into three equal parts. This is very worth noting." (417)

It seems natural to conclude from this, as one modern commentator does, that:

"It is a remarkable tribute to the wisdom of the Greek geometers that two of the problems they highlighted, the duplication of the cube and the trisection of the angle, provide precisely the tools needed for the solution of cubic and quartic equations." (Hartshorne, Geometry: Euclid and Beyond, 271)

Remarkably, Viete's opinion seems to be virtually the opposite:

"The ancients believed that since quadratic equations could be entirely purged [of affection] by such a reduction as this and easily solved, the higher powers could also be completely purged, and they tried to derive from the standard resolution of pure powers a wholly mechanical operation [for doing so] and were so obstinate about it that they did not inquire into any other method for solving affected equations. Thus they tortured themselves hopelessly and spent many good hours at the expense of the mathematics they were cultivating." (240)

I find this a puzzling statement. Obviously the Greeks did "solve" cubic equations, including "affected" ones, i.e., irreducible cubic equations with a linear term, since the trisection of an angle comes down to this. I found this so puzzling that I looked up the original Latin in Viete's Opera, but the translation seems accurate.

I can only infer that Viete's critique is that these solutions do not translate into a recipe for finding numerical solutions. In the case of quadratic equations, one can always, by completing the square, reduce the equation to a "pure power" form, x^2=a. This not only yields a geometric construction of the solutions, but also leads to a recipe for finding the solutions numerically, in the form x=f+sqrt(g) where f and g are polynomial functions of the coefficients of the original equation. Viete seems to be saying that the Greeks sought something analogous for cubic equations, a general method of "completing the cube" which would yield a formula for the solutions of the form x=f+(g)^1/3. But of course there is no such general formula.

To my knowledge there is no direct evidence in Greek sources that they were stubbornly looking for a general solution to cubic equations in this form. But it is true that they offered a great multiplicity of methods for solving third-degree problems, and, I suppose, that these are not numerically interpretable as solution formulas in the manner of Euclid's treatment of quadratic equations. Perhaps, as Viete seems to think, the multiplicity of methods proposed by the Greeks reflect their dissatisfaction or failure in the latter regard.
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