In ancient times, the square was a symbol of human endeavor while a circle represented godly things. Ancient mathematicians therefore worked at devising a formula for "squaring a circle" that would allow them to create a square whose perimeter measured exactly the same as the perimeter of a given circle. The idea was to use these forms as a way of honoring the deities.
Another attempt to square the circle for mathematicians was to create a formula whereby you could create a square whose area was exactly the same as the area of a given circle. Professor Paul Calter of Dartmouth college explains that some scholars think that the pyramids of Egypt was an effort to square the pyramids by creating a tomb whose perimeter at the base equalled the circumference of a circle whose radius is the same as the height of the pyramid.
In 1822 German Mathematician Ferdinand von Lindermann devised a theorem disproving the possibility that a circle could be squared in a finite number of steps using a straight edge and a compass. Since that time, the expression "Squaring the Circle" has gradually become an idiomatic way of expressing the impossible.
In modern English usage, characterizing a plan or effort as an attempt to "square the circle" is a comment on the plan's futility. It can also be used as a shorthand criticism for a flawed theory or argument--one that tries to "square the circle" by glossing over the factors that render it invalid.