What Does Squaring the Circle Mean?

The squared circle
Catherine Beyer

In Euclidean geometry, squaring the circle was a long-standing mathematical puzzle that was proved impossible in the 19th century. The term also has been used as a symbol in alchemy, particularly in the 17th century, and it has a metaphorical meaning: attempting anything that seems impossible.

Mathematics and Geometry

According to mathematicians, "squaring the circle" means to construct for a given circle a square with the same area as the circle. The trick is to do so using only a compass and a straightedge. The devil is in the details:

First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side [square root of] A clearly has the same area. Secondly, we are not saying that [it] is impossible, since it is possible, but not under the restriction of using only a straightedge and compass.

Meaning in Alchemy

A symbol of a circle within a square within a triangle within a larger circle began to be used in the 17th century to represent alchemy and the philosopher's stone, which is the ultimate goal of alchemy. The philosopher's stone, which was sought for centuries, was an imaginary substance that alchemists believed would change any base metal into silver or gold.

There are illustrations that include a squaring the circle design, such as one in Michael Maier’s book "Atalanta Fugiens," first published in 1617. Here a man is using a compass to draw a circle around a circle within a square within a triangle. Within the smaller circle are a man and a woman, the two halves of our nature that are supposedly brought together through alchemy.

Philosophical Meaning

Philosophically and spiritually, to square the circle means to see equally in four directions—up, down, in, and out—and to be whole, complete, and free.

Circles often represent the spiritual because they are infinite—they have no end. The square is often a symbol of the material because of the number of physical things that come in fours, such as four seasons, four directions, and the four physical elements—earth, air, fire, and water, according to ancient Greek philosopher Empedocles—not to mention its solid appearance.

The union of man and woman in alchemy is a merging of spiritual and physical natures. The triangle is then a symbol of the resulting union of body, mind, and soul.

In the 17th century, squaring the circle had not yet been proved impossible. However, it was a puzzle no one had been known to solve. Alchemy was viewed very similarly: It was something few if any had ever fully completed. The study of alchemy was as much about the journey as the goal, as no one might ever actually forge a philosopher’s stone.

Metaphorical Meaning

The fact that no one was ever able to square the circle explains its use as a metaphor, meaning to attempt to complete a seemingly impossible task, such as finding world peace. It is different from the metaphor of attempting to fit a square peg into a round hole, which implies two things are inherently incompatible.