The concept of a cubic equation was entirely unknown in antiquity. Yet:

The famous problem of doubling the cube concerned the insertion of two mean proportionals between two arbitrarily given lengths, an approach introduced by Hippocrates of Chios (470–410 BC). From a modern perspective, this amounts to solving a cubic equation, more precisely a pure cubic equation. A mixed cubic equation arises in a problem presented by Archimedes: the division of a sphere by a plane section. Archimedes derives an equation for a proportion. This equation again can be transformed into a cubic equation. A third type of a cubic equation emerges from the trisection of an angle. However, this insight was not fully recognized until François Viète (1593). Viète explicitly stated that the methods enabling the solution of the Delian problem and the trisection of an angle are sufficient to solve all cubic equations. In doing so, he drew upon the ancient method of neusis, which he legitimized through a postulate of his own. In demonstrating the application of this approach, however, he just gave an example.

The absence of solutions to cubic problems in Archimedes’ work prompted Eutocius, in his commentary, to present a series of solutions to the Delian problem, twelve in total. The first of these solutions is attributed to Plato, the philosopher. In this construction, the endpoints of an L-shaped line are connected by a broken line consisting of two consecutive right-angle turns.This Platonic approach to solving the Delian problem might also be applied to the other problems handed down from antiquity: the trisection of an angle, the construction of a regular heptagon or a nonagon, and the problem of dividing a sphere. Each of these problems can be traced to a cubic equation, by means of a triple proportion. In every case, the solution can then be obtained by a broken line segment “in the Platonian manner”. The incorporation of this method into ancient geometry (in the Euclidean sense) might be achieved by a postulate modeled after Viète’s neusis-postulate, as a supplementum geometriae.

Like the classical method of neusis, the broken-line construction is concerned with the determination of an inclination. The “Platonic Postulate,” however, surpasses all other postulates considered in this context.

Following Viète, Newton likewise states that the general solution of cubic equations is covered by a neusis postulate. Yet in both cases the method was worked out only for particular examples. The fundamental difficulty lies in the restriction to lengths, that is, to positive quantities. This requires a distinction of many separate cases (Omar Khayyam 19; Cardano 18; Viète 4; Newton 8). In the method employing right-angled broken line segments, negative values are included, both for the coefficients and for the solutions, in a natural manner, by the direction the segments are laid off. As a result, the description of the configuration determining the required length (the “construction”) becomes remarkably transparent, even in the broadest sense.

Finally: The method “in the tradition of Plato” can be extended to equations of any higher degree. In such cases, only the number of bends in the broken-line segments increases.