Squaring the Circle

Podcast Transcript

Ancient mathematics was very different than the mathematics you are used to today.

Two primary tools ancient mathematicians used were the compass and the straightedge. With these two simple objects, they made an astounding number of proofs and mathematical discoveries.

However, there were some problems that were always beyond their grasp.

Learn more about squaring the circle and the problem that eluded mathematicians for over 2000 years on this episode of Everything Everywhere Daily.

Ancient Greek mathematicians were some of the first in world history to formalize mathematical thinking.

Numbers and simple arithmetic existed before the Greeks, but the one thing that the Greeks gave to the world was the concept of mathematical proof.

Proofs are a sequence of logical arguments that derive a conclusion from accepted premises or axioms.

Before formal proofs were established, early civilizations like the Babylonians and Egyptians used empirical methods to solve practical mathematical problems.

They employed arithmetic rules and geometric procedures for practical tasks like construction, land measurement, and trade. Their approach to mathematics was more intuitive and lacked formal deductive reasoning.

The Egyptians used simple formulas for the areas and volumes of shapes, but these were based on approximation and observation rather than formal proof.

The Babylonians made advancements in algebra and geometry, particularly in solving quadratic equations and understanding Pythagorean triples, but they didn’t yet employ deductive reasoning or proofs.

So, while the Greeks created the idea of “ the proof,” which is the foundation of mathematics, their mathematics was a far cry from what we have today.

For starters, they didn’t use base ten numbers like we do today. Their numbers were more like Roman numerals, which are notoriously difficult to manipulate.

They didn’t have zero or negative numbers, which made things even more difficult.

Their mathematics wasn’t really about number manipulation or calculation at all. They didn’t have algebra, which Islamic mathematicians would invent centuries later.

What the Greeks really did excel at was geometry.

If you have ever taken a geometry course at any point in your life, there is a very good chance that what you studied came from Euclid 2,300 years ago.

Their geometry was abstract, but it also had a very real, tangible element to it.

The Greeks used two very simple tools to do most of their geometry work: the compass and the straightedge.

You probably have used both of these tools at some point in your life.

A straightedge is, just as the name implies, a straight edge. You probably use a ruler, but the actual measurement marks on the ruler are irreverent to a straightedge.

A compass is a tool for drawing circles or arcs. It typically consists of two hinged legs, one with a pointed end and the other with a pencil or marking device.

With just these two things, you can create an incredible number of shapes and prove theorems.

Constructing regular polygons such as equilateral triangles, squares, and pentagons.
Bisecting angles and constructing perpendicular lines.
Calculating areas and volumes of simple shapes.

Within this system, compass and straightedge constructions were not just a mathematical exercise but a form of logical rigor.

These constructions represented “pure” geometry, free from the complications of measurements or instruments beyond the idealized straightedge and compass.

While the Greeks were able to do quite a bit with these simple tools, they couldn’t do everything. In fact, there were a few problems that absolutely confounded the Greeks.

In particular, there were three that no one could find a solution to using a compass and straightedge:

Doubling the Cube, Trisecting the Angle, and Squaring the Circle.

All of these problems are relatively simple to state but were fiendishly difficult to solve.

The legend behind the Doubling the Cube problem, also known as the Delian problem, originates from ancient Greek mythology and centers on the island of Delos. According to the story, the people of Delos were suffering from a terrible plague and sought advice from the Oracle of Delphi.

The oracle conveyed a message from the god Apollo, stating that to end the plague, they must double the size of his cubic altar. Taking the instruction literally, the Delians interpreted this to mean they had to double the volume of the altar, which was in the shape of a cube.

The problem then became how to construct a cube with twice the volume of the original altar, using only the basic geometric tools of the time: a compass and straightedge. Despite their best efforts, the ancient Greeks struggled to find a solution, as simply doubling the side length of the cube would result in a cube eight times the volume.

This challenge was passed down through generations of mathematicians, who attempted to solve it as a geometric puzzle. The legend highlights both the reverence the Greeks held for geometry and their belief in divine guidance to solve practical and spiritual problems. Ultimately, it was not until the 19th century that the problem was proven impossible to solve using classical geometric methods.

The trisecting the angle problem is a classical challenge in geometry that involves dividing an arbitrary angle into three equal parts (or trisecting it) using only a compass and straightedge.

Bisecting an angle had been solved in antiquity.

While it is possible to trisect certain specific angles, such as a right angle (90°), the general problem of trisecting any given angle could not be solved with just these tools.

Mathematicians like Hippias and Archimedes explored mechanical methods to approach the problem, such as using curves or other tools beyond the compass and straightedge, but the Greeks were unable to solve it within the constraints of their geometric rules.

The impossibility of trisecting an arbitrary angle was formally proven in the 19th century when it was shown that certain angle divisions require solving cubic equations, which cannot be done with just a compass and straightedge.

This is because only numbers that are solutions to certain types of quadratic equations (associated with the algebraic constructibility of lengths) can be constructed using these classical tools, and cubic equations like those arising from angle trisection fall outside this range.

Finally, we get to the problem, which is the most famous of the three and has become a metaphor for unsolvable problems: the squaring of the circle.

The challenge is deceptively simple: construct a square with the same area as a given circle using a compass and straightedge in only a finite number of steps.

That sounds really easy. It is anything but.

One of the earliest recorded figures to attempt squaring the circle was the Greek philosopher Anaxagoras. In the early 5th century BC, he worked on this problem while imprisoned, making him one of the first known mathematicians to grapple with the challenge. Anaxagoras’ work was based on intuition and approximations, and although he didn’t solve the problem, his effort illustrates the problem’s appeal to ancient thinkers.

In the late 5th century BC, Hippocrates of Chios made significant progress by discovering that certain curved shapes, known as lunes, moon-shaped figures bounded by arcs, could be squared. His work showed that it was possible to square some segments of a circle, raising hopes that the full circle could be squared. Hippocrates’ discovery was important for later studies, but ultimately, his method could not be extended to the whole circle.

Around the same time, Antiphon, a contemporary of Hippocrates, suggested an interesting approach by inscribing regular polygons inside a circle. He hypothesized that by increasing the number of sides of the polygon, it would eventually coincide with the circle, thus allowing one to approximate the area of the circle.

If this sounds familiar, he developed an early form of what would eventually become integral calculus.

In the 3rd century BC, Euclid published his monumental work, The Elements, which systematized the rules of geometry. Euclid outlined a rigorous system of geometric constructions that relied on just a compass and straightedge, defining the rules that would constrain all future attempts to square the circle.

Euclid could not provide a solution to the problem but solidified the importance of proving geometric constructions through logical deduction rather than trial and error. The Elements became the standard reference for geometric work for over 2,000 years, and squaring the circle remained an open challenge within this framework.

During the Middle Ages, particularly in the Islamic Golden Age from the 8th to 14th centuries, mathematicians preserved Greek mathematical texts and expanded on them. Although there were no major breakthroughs in solving the problem of squaring the circle, Islamic mathematicians such as Al-Khwarizmi and Omar Khayyam made important contributions to algebra and geometry, setting the stage for future developments.

Some mathematicians explored more sophisticated geometric methods and improved approximations of ?, but they did not resolve the fundamental problem. The work of Islamic scholars was transmitted to Europe during the Renaissance, where it reignited interest in classical problems like squaring the circle.

At this point, mathemiticians realized the issue was really not with any technique with a compass and straight edge, it had to do with the number pi.

You can easily create an equation showing the area of a square equaling the area of a circle.

The area of a square is s squared, where s is one side. The area of a circle is pi r-squared, where r is the radius.

Set them equal to each other and divide each side by r-squared and you get s-squared over r-squared equals pi.

The secret to solving the puzzle was figuring out the nature of pi.

Throughout the Renaissance, mathematicians attacked the problem by trying to figure out the nature of pi. Much of this involved calculating pi to more and more digits. While that was helpful, it didn’t really tell you anything about the nature of pi.

One big breakthrough that would be a big step towards resolving the problem was taken by Godfried Leibniz in 1682 who found the function sin x was not algebraic. That means you couldn’t solve it by using algebraic operations such as addition, subtraction, multiplication and division.

This was the first case of the proof that some things could not be solved algebraically.

Numbers that couldn’t be expressed algebraically were called transcendental numbers.

In 1768, Johann Heinrich Lambert proposed but did not prove, that pi and e were both transcendental numbers. This hypothesis was important because if it was true, that would mean that squaring the circle was in fact, impossible. Lambert also proved that pi was at least an irritation number, meaning it couldn’t be expressed as a fraction using whole numbers.

After almost 2000 years of fruitless attempts at squaring the circle, the decisive breakthrough came in 1882, when German mathematician Ferdinand von Lindemann proved that ? is a transcendental number.

This result was built upon earlier work by the French mathematician Charles Hermite, who had shown that the number e, the base of natural logarithms, was transcendental.

This was the final straw. After 2000 years of trying, it turns out that everyone who was trying to square the circle was wasting their time, because it was impossible to do.

The centuries of pursuit of trying to square the circle ended up being futile, but it wasn’t a total waste. In the process, there were great deal that was learned about geometry, algebra, and the number pi.

In the end, despite being a relatively simple problem to state, squaring the circle was shown to be impossible.

The ultimate lesson was that some problems, no matter how simple they seem, can never be solved.