How was geometry historically used to solve polynomial equations?

Question

I'm researching historical use of geometry to find solutions to polynomial equations. I'd like to ask for those familiar with this topic, could you describe the use of geometry by early mathematicians to solve polynomial equations? Could you also offer some examples to illustrate the ideas?

Answer 1

Modern style polynomials were introduced in 12th century by Arabic mathematician Al-Samawal, so before that we can only talk about problems that reduce to polynomial equations, some obviously, some not so much. Many such problems came from geometry, so naturally they were solved using geometry.

Quadratic "equations", i.e. geometric problems today convertible into them, were solved by Sumerians and Egyptians as early as 2000 BC. Some computational prescriptions for doing it, like false position, survived, see The origin of quadratic equation in actual practice. Pythagoreans did it by literally completing squares with straightedge and compass, Euclid describes it in book II of Elements. This used to be mischaracterized as "geometric algebra".

The famous Delphic problem of duplicating the volume of a cube shaped altar reduces to solving $x^3=2$. One of the first solutions was given by Menaechmus, a friend of Plato's, by intersecting two conic sections. In modern notation they are two parabolas $x^2=y$ and $y^2=2x$, or parabola and hyperbola $x^2=y$ and $xy=2$. In On Sphere and Cylinder Archimedes posed a problem of cutting a sphere in a given ratio, which reduces to $x^2-x^3=s$, where $s$ is a parameter, and solved it also by intersecting conic sections.

This method was perfected by famous Persian poet and mathematician Omar Khayyám. Unlike the predecessors who started from a geometric problem and proceeded geometrically, he began with an algebraic classification of cubics, and only then turned to geometry and constructed conic section solutions for all 14 non-trivial cases. For example, Archimedes's equation is of the type $x^3+a^3=bx^2$ in Khayyám's classification, and (modernizing) he solves it by intersecting the hyperbola $xy=a^2$ with the circle $x^2+y^2=bx$. Netz's Transformation of Mathematics in the Early Mediterranean World is a good source on these topics, its subtitle is From Problems to Equations.

The problem of inscribing regular polygons into a circle also reduces to solving a polynomial equation, although that was discovered much later, namely finding complex roots of $z^n=1$. For $n=3,4,5,6,8,9,10$ it was solved with straightedge and compass already by Pythagoreans, which is described in book IV of Euclid's Elements. For $n=7$ Archimedes gave a solution with compass and marked straightedge using a construction known as neusis or verging. Aaboe's Episodes from the Early History of Mathematics gives a nice exposition.

Inscribing regular polygons is a particular case of $n$-secting an angle into equal parts, which corresponds to solving $z^n=a$, where $a$ is a complex number of absolute value $1$. Trisection with $n=3$ is another famous ancient problem, which is known to be unsolvable with straightedge and compass, but a neusis solution was also given by Archimedes. Baragar's Constructions Using a Compass and Twice-Notched Straightedge gives a modern perspective on neusis, and its relation to algebraic equations. Other ancient solutions used intersections of conic sections (Apollonius), and mechanically generated curves (conchoid of Nicomedes and cissoid of Diocles). Other curves, quadratrix of Hippias and Archimedean spiral, can be used to $n$-sect any angle.

P.S. Ironically, Khayyám was dissatisfied with his geometric solutions despite perfecting them. He wrote in Al-jabr w'al-Muqabala: "When however the object of the problem is an absolute number neither we nor any of those who are concerned with algebra have been able to solve this equation - perhaps others who follow us will be able to fill the gap..." Others tried in vain, and in 1494 Summa Arithmeticae Luca Pacioli went so far as to say that algebraically "solving the cubic is as impossible as the quadrature". Only 19 years later Scipione del Ferro algebraically solved the first cubic.