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.
FAQs
“The 'squared circle' or 'squaring the circle' is a 17th century alchemical glyph or symbol for the creation of the Philosopher's Stone. The Philosopher's Stone was supposed to be able to transmute base metals into gold and perhaps be an elixir of life.”
Has squaring the circle been solved? ›
Despite the proof that it is impossible, attempts to square the circle have been common in pseudomathematics (i.e. the work of mathematical cranks). The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible.
What is the geometric problem of squaring a circle? ›
Construct a square equal in area to a circle using only a straightedge and compass. This was one of the three geometric problems of antiquity, and was perhaps first attempted by Anaxagoras. It was finally proved to be an impossible problem when pi was proven to be transcendental by Lindemann in 1882.
What is the expression squaring the circle? ›
Definition of 'square the circle'
If you say that someone squares the circle, you mean that they bring together two things which are normally thought to be so different that they cannot exist together.
What does the square mean in alchemy? ›
The squared circle: an alchemical symbol (17th century) illustrating the interplay of the four elements of matter symbolising the philosopher's stone.
What do the circles mean in alchemy? ›
In the field of Alchemy, Alchemical Circles are an aid to magic that alchemists have discovered. It uses the memetic properties of magic in general in order to make it easier to perform certain feats of magic, make them more powerful, or even possible at all.
Has anyone squared a circle? ›
The proof that the circle cannot be squared came in 1882, when Ferdinand von Lindemann proved that pi is transcendental, meaning that it is not the root of any polynomial.
What is the hardest circle theorem? ›
Alternate Segment Theorem: The angle between the tangent and the side of the triangle is equal to the opposite interior angle. The angle between the tangent and the triangle will be equal to the angle in the alternate segment. (This is the hardest rule and can be tricky to spot).
Can a circle be geometric? ›
Lines and circles are the most elementary figures of geometry − a line is the locus of a point moving in a constant direction, and a circle is the locus of a point moving at a constant distance from some fixed point − and all our constructions are done by drawing lines with a straight edge and circles with compasses.
What is the Lune of Hippocrates? ›
In geometry, the lune of Hippocrates, named after Hippocrates of Chios, is a lune bounded by arcs of two circles, the smaller of which has as its diameter a chord spanning a right angle on the larger circle.
A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre.
Can God create a square circle? ›
Omnipotence, they say, does not mean that God can do anything at all but, rather, that he can do anything that is possible according to his nature. The distinction is important. God cannot perform logical absurdities; he cannot, for instance, make a square circle.
What is the symbol for squaring? ›
What is x²? An exponent of 2 is the mathematical symbol for squaring a quantity, i.e., multiplying that quantity by itself. Thus, x² means that x is being multiplied by itself.
What is the symbol for squaring something? ›
The exponent two (2) is the same as squared (^2), so looking at the equation e^2 can help you remember that this means to square the number.
How is the missing square puzzle possible? ›
The key to the puzzle is the fact that neither of the 13×5 "triangles" is truly a triangle, nor would either truly be 13x5 if it were, because what appears to be the hypotenuse is bent. In other words, the "hypotenuse" does not maintain a consistent slope, even though it may appear that way to the human eye.
Are we going to solve for the area of a circle? ›
The area and circumference of a circle can be calculated using the following formulas. Circumference = 2πr ; Area = πr2. The circumference of the circle can be taken as π times the diameter of the circle. And the area of the circle is π times the square of the radius of the circle.
