Speculations on the Circle and the Square
W. Bro Michel Bourassa,
IPM Yellowknife #162 GRA
Victoria-Columbia No. 1 BCR

From the beginning, 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 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. As a result, even today, the term “squaring the circle” has come to mean a vain or impossible activity.

Yet the answer seems apparent and simple: calculate the circumference of the circle and divide by four and multiply. This is deceptively simple, and ultimately, of course, as 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 area no matter how hard we try. The circumference cannot be evenly divided , thus the area 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 B  or pi.

The value of B  in the formula Area =Br2 will therefore give us only an numeric approximation of the area of the circle and in fact we will never be able to compute it’s exact area. It’s 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 man.

From ancient times to today, circles have mystical meanings. For the ancients it 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, encompassing, 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 and rework 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. It began, and is today, a guide to every man who is making the attempt to resolve that quest.

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 laboring 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, 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 compasses.

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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, 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.

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 scope 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. 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 labor we will be better than yesterday. We are, in effect, attempting to “square the circle” not though mathematics but by conceptually transforming the spherical form of the sky and the heavens into the rectangular form of the earth, bringing heaven to earth. A life’s labour will thus merge the Devine with the earthly.

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.

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 form of the square.)

© 2006 Michel Bourassa

Sources:
Jones, B. E. Freemasons’ Guide and Compendium
Lawlor, R. Sacred Geometry
J.C. Cooper, Encyclopedia of Symbols
Various anonymous Internet comments

Steilacoom Lodge No. 2
A Proud Founding member of the Most  Worshipful Grand Lodge of
Free and Accepted Masons of Washington