As Masons, we are all introduced during our ritual lectures to the Masonic symbol of the Point Within a Circle, and instructed in its’ allusion.  The most interesting thing to me during my own such introduction was that the figure representing this symbol contained not only a point within a circle, but also two straight vertical lines touching the sides of the circle. It was explained during the ensuing lecture that these lines represented the two Holy Saints John, namely John the Baptist, and John the Evangelist. This struck me as peculiar to say the least, and I have been trying to figure out this peculiarity ever since.

 In the course of my inquiry I found several explanations, including one[i] which pointed out that the Feasts of the two Saint Johns are separated by six months time, and that the symbol of the Point Within a Circle is a sort of miniature ornery showing the path of the earth about the sun, with the feasts designating winter and summer solstices[ii].  Another explanation likened the circle to an astronomical or astrological diagram, complete with astrological symbols arranged about the circle circumference, and which held that the vertical lines were representative of the tropic of Cancer and the Tropic of Capricorn[iii]. Yet another variation[iv] of the explanation of the Point Within a Circle also identified the vertical lines as signifying the two Saints John, but expounded upon the significance of the VSL in the symbol and offered an exhaustive discussion of chapters and verses within the Bible attributed to Saint John the Baptist and St. John the Evangelist all of which alluded to the Point Within a Circle representing God and man, respectively. A further version was discovered which ignored the vertical lines, but which asserted that the Pint Within a Circle was the Monad[v], and represented God.

These various explanations, though they were all plausible in the world of Masonic symbolism, did not satisfy my curiosity. I began to consider that the figure representing the point within a circle is reminiscent of a drawing which one find in a textbook illustrating some principle of Geometry. We are all acquainted with the Masonic symbol of the 47^th Proposition of Euclid; I began to wonder if the point within a circle might be a similar construction.

It was at about this time that through further reading I discovered what was described as a closely held secret[vi] of ancient craft Masons; namely that if one is to draw a circle and then draw a further line across that circle through its’ center point (marking it’s diameter) a right triangle can be simply yet consistently constructed. The technique involved is to draw a line starting at the point where the line through the circle center intersects the circle circumference (Point A in Figure 2), and to extend that line until it touches the circle at any point on its’ circumference (Figure 2, Point C). The line is then continued from the point of intersection with the circumference to the point at which the center line intersects the circle on the opposite side of the circles’ diameter (Figure 2, Point B) and is further continued to the start (Figure 2, Point A).  The end result is that a right triangle is constructed regardless of the point on the circles’ circumference selected. Try it yourself, it works very nicely.

This method for the construction of a right angle is presented as Theorem 12, in Book III of Euclids’ Elements[vii]  (Euclid, III. 3.), “An angle inscribed in a semi-circle is a right angle”.  Euclid wrote: “Let angle ABC be inscribed in the semi-circle ABC; that is, let AC be a diameter and let the vertex B lie on the circumference; then angle ABC is a right angle”.  Although presented in Euclids’ Elements and provided with a proof formulated by Euclid, it was the Greek Philosopher, Astronomer, and Mathematician Thales of Miletus[viii] (ca. 624 BC–ca. 546 BC), who is credited with the first publication of this theorem.  

Naturally, as a trade Secret this technique would have proven extremely valuable to the ancient Craft Masons, and could be used among other things, to check the squares of workmen to ensure that they were true. . It is also probable that the development of scientific surveying and navigating instruments such as the astrolabe made use of this, or a similar Theorem in their construction and as their operating principle.

When reading this I recalled the symbol of the Point Within a Circle, and began to wonder if the Point Within a Circle might actually be a diagram which was intended to be used as a proof of Thales’ Theorem. I decided to research the matter and in the course of doing so, discovered an excerpt from a Masonic Handbook[ix], which (regarding the point within a circle) stated: 

“Ritualistically, this is a symbol of control of conduct; a standard of right living. The symbol has an extreme antiquity. Early Egyptian monuments are carved with the Alpha and the Omega or symbol of God in the center of a circle embordered by two upright parallel perpendicular serpents, representing the power and wisdom of the Creator. The symbol apparently came into Masonry from an operative practice, known to but a few Master workmen on Cathedrals and great buildings.

Any school boy knows it now; put a dot on a circle anywhere; draw a straight line across the circle through its center; connect the dot with the points at which the line through the center cuts the circle; the result is a perfect square.

This was the Operative Master’s great secret - knowing how to “try the square” It was by this that he tested the working tools of the Fellows of the Craft; did he do so often enough, it was impossible either for their tools or their work “to materially err”.

I nearly considered the matter settled, and was convinced that I had drawn the correct conclusion thinking that the Point Within a Circle was a clue to the Geometric construction by which Thales’ Theorem could be derived.  With further reflection I decided to confirm this by looking at the published proof offered by Euclid.  I was surprised to discover that the proof of the theorem did not involve the construction of parallel tangent lines at all, but rather relied upon the construction of a simple radius to the vertex producing the right angle[x] (see Figure 3). Further investigation revealed that over the centuries there have been numerous other proofs of the Theorem developed, all different from that of Euclid, but none of them employing two parallel tangent lines.

            I quickly began to suspect that Thales’ Theorem was not actually related to the Point Within a Circle, but rather that the Point Within a Circle was associated with a different Geometric construction. I combed Geometry Textbooks, Masonic Encyclopedias, and the Internet searching for a Geometric figure which was constructed using a circle, a point at the center of the circle, and two parallel tangent lines on that circle. What I discovered was a second method by which a right triangle could be constructed, using these very elements along with one additional straight line, which is also a tangent to the circle. I have reproduced this figure as described by my source[xi] (Figure 4) below, along with an explanation of how the figure is drawn.

In Figure 4, AB and CD are two parallel tangents to a circle having a center O. A random tangent EF with point of contact G intersects AB at point H and CD at point I. Beginning with the center of the circle O a line is drawn connecting O with point H. The line is extended to connect point H to point I, and continued back to the center of the circle, Point O.  This construction produces a triangle in which angle HOI is 90^o. 

A narrative description of this technique is that given a circle with two parallel tangents, and a third tangent drawn randomly to the circle which intersects both of the two parallel tangents, a right angle may be constructed by connecting the center point of the circle to the points at which the tangents intersect. The proof of this construction is similar to that used for Thales Theorem (see footnote 9) and requires construction of a segment from the circle center O to the tangent point G.

It is, I believe quite amazing that the elements of this construction so neatly utilize the framework contained within the Masonic symbol of a Point Within a Circle.  This discovery however leads to other questions. For example, why was this particular method for the construction of a right triangle used as a Craft symbol, when a simpler method (Thales Theorem) existed, and was chronicled by Euclid ?  If this construction is indeed the intended functional use of the Point Within a Circle symbol, it is probably of some importance to the Craft; yet this construction as a Geometric function is fairly obscure and is not considered by Euclid as a Theorem or Proposition (although several of Euclids’ propositions are used to establish the proof). Answers to these questions remain to be discovered, and I intend to find great entertainment in further exploration. A friend of mine once said “Most of what I know I learned while looking up the answer to something else”. The truth of that humorous comment is rarely more evident than when examining the ancient symbols of our Craft.

Since my publication of my article “The Point Within a Circle – More than Just an Allusion ? ” I have been contacted by a fairly large number of Brethren who have requested further details concerning the method which I described for constructing a right triangle. The method, uses as its basis the Masonic Symbol of “The Point Within A Circle” (complete with its two parallel lines tangent to the circle) to construct a right triangle.

As mentioned in the article, this method relies upon the Theorem of Thales.  This Theorem instructs that if you draw a circle with its center marked, and then construct a straight line through the center of that circle, intersecting the circle at two points on its circumference, you have drawn the circle’s diameter) and that if you now choose any additional point on that circles circumference and connect the ends of the line forming the diameter to that point with straight lines, you will have drawn a right triangle[1]

In the method within the article I began by selecting a point on the circle’s circumference, and then constructed a line tangent to the circle at that point. I produced a right triangle by connecting the center of the circle to the points at which the tangent line intersected the two (vertical) parallel lines contained within the Symbol of the “Point Within A Circle”. 

Since there have been so many inquiries as to how and why this works, I have constructed a diagram (Figure 2) which illustrates the method. Consider the symbol of “The Point Within a Circle” with the center of the circle identified as Point “O” and in which I have constructed Line “A-B” to mark the circle’s diameter.  I choose a random point on the circumference of the circle (Point “E”) and construct line “O-E which joins the center of the circle with point “E” which lies on the circle’s circumference”.  I now draw a line perpendicular to Point “E”, establishing a tangent line to that point. Note that the tangent line just constructed intersects both of the Parallel Lines of the original figure of “The Point Within A Circle” at points “C” and “D” respectively. By joining the two intersecting points of the tangent line with the center of the circle, I form Triangle “COD”, which is a right triangle.

Why does this work ? The answer becomes apparent if I inscribe a new circle with Center “F” and having diameter “C-D” (or radius “C-F”) around the triangle. The center point of the original circle (Point “O”) lies on the circumference of the new one, and triangle “COD” can be shown to be right in accordance with the Theorem of Thales.

 I hope this illustration and accompanying explanation will clarify my use of this technique in the original article. 

As a parting aside, I would like to note that intersecting circles with their center points identified and two parallel tangent lines (i.e. “The Point Within A Circle”) have fairly broad application for the Geometer, beyond the construction of right triangles. Not surprisingly, many of these applications are of significant Masonic interest. For example, the Vesica Pisces, an ancient symbol well known to Freemasons is constructed using two overlapping circles of the same diameter (see Figure 3) in which the circumference of each circle intersects the center of the other.

 In Figure 3, the Vesica Pisces thus constructed is outlined using a blue cross-hatched pattern. Beyond the classic symbolic meanings of the Vesica Pisces, the symbol is closely related to the so called “Divine Proportion”, “Golden Ratio”, or simply the Greek symbol Phi[2] (f). Phi, as any fan of Dan Brown novels will realize is associated with a wide range of naturally occurring phenomenon and objects, including the pattern upon which plant leaves develop and upon which natural spirals develop in seashells and even the Milky Way galaxy.  It is considered an irrational number, meaning that the exact numerical value of Phi cannot be determined because it’s calculation creates an infinite series of non – repetitive digits. In Figure 3, the line segment on the triangles hypotenuse “I-K” is equal to  which is Phi, or 1.618. The Hypotenuse of the triangle is equal to  or 1.732. For a more complete discussion of the geometric properties of the Vesica Pisces the reader is referred to an excellent article in the Nexus Network Journal by Rachel Fletcher, entitled “Musings on the Vesica Piscis” [3]. For a more mathematically oriented discussion of the Divine Proportion coupled with a bit of Philosophy, I recommend another article from the Nexus Network Journal by Scott Olsen [4].