Speculations on the Circle and the Square

shared with the permission of

W.Bro Michel Bourassa, PM Yellowknife #162 GRA &

Victoria-Columbia #1  GLofBC&Y


Speculation: Origin: 1325–75; Middle English speculacioun < Late Latin specul?ti?n- (stem of specul?ti?) exploration, observation. See speculate; the contemplation or consideration of some subject: to engage in speculation on humanity’s ultimate destiny.


From the beginning of thought, philosophers, mathematicians, thinkers and our operative predecessors have struggled at length and without success with a seemingly simple problem: Using but a compass and 24 inch gauge construct a square that has the exact same area of a given circle. So simple at first glance, yet so confounding in application. In fact, a solution to the problem has eluded them all and ultimately proven to be impossible even today.  Thus, in modern culture and language use, the term “squaring the circle” has come to mean a vain or impossible activity.


At first glance the answer seems apparent and simple: measure the circle’s circumference, divide by four and multiply and we have an area computation. This is deceptively simple, and ultimately, of course, a deception – it simply will not provide the answer. Let me demonstrate:


Proceeding with that approach, and giving a value of “1” to the diameter of the circle we find that we can only approximate the circle’s circumference no matter how hard we try. It will always be of the incommensurable; “irrational” number type, 3.14159………which can be carried out to an infinite number of decimal points without ever arriving at a resolution. This number we know by the Greek symbol  pi.


The value of Pi in the formula Area =pi R2 will therefore give us only a numeric approximation of the area of the circle and thus we will never be able to compute its exact area. It is conceptually amazing really: draw a circle, large or small, and one cannot compute its area! It will always be indefinite. Not knowing the exact area of the circle we therefore cannot calculate the exact sides or area of our square. It follows therefore that a square’s dimensions and the area it contains can only be an approximation of that of a circle.


Why did ancient philosophers and thinkers spend so much time in trying to devise ways of squaring the circle? It had to do with reconciling geometry, measuring the heavens, earth and ultimately, man.


From ancient times to today, circles have mystical meanings. For the ancients a circle represented many things but central to its meaning was that it represented the pure unmanifest spirit-space, the heavens, unity and the infinite. In esoteric astrology it represents the creative spark of divine consciousness that exists in every individual; the desire to live and an individuals’ life energy.

The measurable square on the other hand represents the manifest, and comprehensible world – the earth.  “The square is associated with earth and matter”. (Daniel Beresniak) It is calculable, encompassable, finite and can be grasped by the mind as well as the hand.


Described in a different way, the Philosophers struggle was this: How to take the infinite mind and spirit and metamorphose it into something real and practical; How to take pure thoughts, morals, and principals, rework and match them into practical useful adjuncts of every day life. Freemasonry, being a beautiful system of morality veiled in allegory and illustrated by symbols, draws from that heritage. Freemasonry began, and is today, a guide to every man who is making the attempt to resolve that quest and perfect himself.


In Freemasonry we find the symbol of a circle with a dot within, a meaning for which is given in the dialogue between the WM and the Wardens while labouring in the 3rd degree. In addition we are urged to use a compass to draw a circle that represents our duty to keep within “due bounds” with all mankind, the circle representing those bounds, and its area indefinite. Indefinite because it is for every man to define his own bounds and how large or small the area inscribed by the application of his compass shall be .


In speculating upon the circle and the Masonic imagery associated with it we see, as the ancient philosophers did, that the circle in Freemasonry implies the extent of one’s duty and place in the universe – a partially defined and incalculable area. It is a dimension that each Mason must draw and define for himself as only he can – its size, and circumference and the area it encompasses. The extent or area of our duty is at first like the circle – incalculable, infinite and abstract. But Freemasonry strives to make what is philosophical and ephemeral into what is practical and real on a day-to-day basis.


At the same time as we study the circle, Freemasonry urges us to consider the ashlars and the square: The lines of a square are measurable precise, sharp and defined. They have been tried and proven. The area contained therein is readily calculable with the 24” gauge.


Thus the circle represents our vision, our reach that is for each of us to define; the square represents our realty, our grasp, and what we actually build and maintain in terms of duty, behavior and conduct. By applying the lessons of the working tools one can move from the intangible symbolism and indefinite bounds of the circle and our ideals to the small day-to-day actions of a moral life, a life forming a square and thereby approximate what philosophy calls forth by the circle. For the heart may conceive and the mind may devise but it is all in vain if the hand is not  prompt to execute  the design. While we may never be able to achieve all that we hold and admire as ideal we can be better men today and tomorrow. We may fall short of the ideal, but with labour we will be better men than yesterday. We are, in effect, attempting to “square the circle” not though mathematics but by conceptually transforming the rectangular form of the earth into the spherical form of the sky and the heavens, bringing heaven to earth.


To think morally, lovingly and generously is “circular”; to actually act morally, lovingly and generously is “square” – and Masonic.

The more one practices with the working tools and adjusts his life and conduct according to their lessons, the higher degree of perfection one attains as a man and when a near equality is drawn between the “circle” and the “square” the individual will be able to express all of his circle’s dimensions or qualities though the finite. The circle with a square within depicts the union of heaven and earth, by analogy, the perfect man. This is consistent with Kabalistic philosophy that the goal of all humans is to perfect the emotional and intellectual, and transcend them, being reborn into the divine and seeking understanding with light.  And isn’t “Light” the  first request of every initiate to Freemasonry?



Sengai, c 1830 Mitsu Art Gallery, Tokyo. (Japanese Zen drawing showing “creation” though a progression from the unity of the circle though the triangle to the manifest forms of the square.)


© 2011 Michel Bourassa




Jones, B. E. Freemasons’ Guide and Compendium

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Various anonymous Internet comments