Rousseau compares putting law over men to squaring the circle in geometry.Can you square a circle? Means what?

Question

Question comes from Jean-Jacques Rousseau's "The Government of Poland". The question arises from paragraph 7 from chapter I, "The Issue Posed". The question is about how to square a circle, or if it is even possible. One major part is that it does not identify the part to be squared, only the circle itself, but I'm sure someone with a mathematical background will be able to easily explain how "Putting law over men" can be related and compared to "the squaring of a circle in geometry." Thanks.

Answer 1

Squaring a circle means trying to draw, with a compass and straightedge, a square with an area equal to the circle that is already drawn.
This is a very well known math problem, and it is an impossible one.
It boils down to trying to estimate the value of √π, a transcendental number. You can approximate it, but not find an exact value.


"Squaring the circle" is a metaphor for something that is hopeless or vain, or something that requires an infinite number of approximations and iterations. You can see how that applies to law.

Answer 2

"Squaring the circle" is an ancient problem in geometry, probably dating back to the Egyptians......

The problem is this; is it possible to construct a square with EXACTLY the same area as any given circle, using only simple tools such as a compass and ruler.

Over the ages, thousands of mathematicians, professional and amateur, have tried their hands at this problem with dubious sucess.

Only, finally in 1880, the mathematician Carl van Lindemann proved that this task was, in fact, impossible.

Now, the area of any circle is

A = π r².

What Lindemann did was to prove that π is "trancendental." In other words, the *exact* value of π can NEVER be calculated using a finite series of algebraic steps.

Now, mathematically, operations using a ruler and compass can be shown to have algebraic equivalents (i.e. operations such addition, subtraction, division, square roots, etc.) Therefore, in order to square a circle *exactly*, it would require an *infinte* number of steps.

Rousseau's work was, of course, a century and a half earlier than the proof by Lindemann. Rousseau wouldn't have known that the problem actually was impossible, only that it was *probably* impossible, since no one had made any progress toward solving it for thousands of years

In this sense, Rousseau uses the phrase "Squaring the Circle" as a metaphor for a goal which is tantalizing, but almost certainly futile.

~W.O.M.B.A.T.

Answer 3

"The quantity (pi) has been calculated to more than 2,000 decimal places, and we know it will never come out evenly. The Greeks did not fully recognize (pi)'s irrationality and so they wasted much labor trying to solve the one big problem which this fact made impossible-constructing a square whose area is equal to that of a given circle, or literally trying to 'square the circle' ".
My additional note: (pi) has now been computed to over a million decimal places. It still remains a non-
terminating, non repetitive number: still irrational.

I can only surmise that Rousseau meant that it is
impossible to impose laws that all men will obey. I
leave the correct interpretation to you.