Vol. XXXI No. 4 — April 1953

“From A Point To A Line. . . .”

To be informed, a Mason should study the relationship between geometry and Masonry. Indeed, the Fellowcraft Degree has much to say on the subject and enjoins all who receive it of the importance of this science of measurement.

The majority of men who studied geometry in school have forgotten its rudiments. Its study in formative days doubtless served the purpose of disciplining young minds and, at least in the hopes of those who arrange educational programs, taught its students to think.

But in the years of manhood, geometry for most of us is nothing but a difficult subject we once studied, and have now forgotten. Freemasonry enjoins a consideration of many geometrical figures and terms. This Bulletin points out but a few of these. Each deserves study for itself, and those whom these pages may interest will find little difficulty in tracing the whole stories of these mathematical symbols in modern Masonic books and encyclopedias.

Here are but sketches made with the hope of interesting those to whom, otherwise, geometry and its figures become even less attractive when they are also Masonic symbols.

Brethren meet the circle in Freemasonry in “a certain point within a circle” and in the rite of circumambulation.

This somewhat ponderous word comes from the Latin, and means, literally “to walk around” as brethren have walked around their altars since the days when the first sun worshippers walked around their stone altars, in humble imitation of the course of the sun — from east to west by way of the south, or, as we would say, clockwise.

It is interesting, and to some rather pitiful, that the circular character of the procession about a holy spot, or Sanctum Sanctorum, should have become practically a square. The square of course is among the principal Masonic symbols. It was not unnatural for brethren interested in formalizing the degrees (making the movements of those taking part as exact as the words they spoke) to become more and more insistent that brethren should “make square corners” in movements about the lodge.

But circumambulation puts the emphasis on the matter of going around, not upon the manner. Its importance is in the thought behind, that man can imitate his Creator, even if in but humble ways. Walking around an altar is a tie with a practice so old no man knoweth when it began — a perambulation that tried to express to a dimly felt Supreme Presence a humility of spirit in this imitation of the suns course, the sun being the only god ancient men knew.

The “point within a circle” has engaged the most devoted of Masonic students over a long period of time and many and various have been the explorations into and the resulting explanations of its importance and how it came into the symbolism of the Ancient Craft. The consensus now is that originally the point in the circle, crossed by one (not two) lines, was a reminder that by its use a perfect square could most easily and accurately be made. How the single line became two, and those two changed from the upright and parallel serpents of ancient Egypt into representations of the Holy Saints John no one has yet satisfactorily determined — unless speculation and argument can be denominated determination.

The point in a circle is far older than any possible Freemasonry. The use of it, the compasses, the straight edge, to form a perfect square is older than Pythagoras or Euclid. The Holy Saints John came into Masonic symbolism but a few hundred years ago.

One of the great satisfactions in Freemasonry is that when a symbol has many interpretations, each may choose that which is most enlightening and understandable to him. The point within the circle, the circle itself, circumambulation, all have several possible and all veritable meanings, each of which appeals to a different variety of mind and knowledge and Masonic experience.

Few geometrical figures in Freemasonry cause more comment than that denominated in rituals as “an oblong square.” To many the term is a self-contradiction, in the same class as “a square circle” or “a four-sided triangle.”

Actually our figure is a parallelogram, two long and two shorter sides and all corners right angles. In a much older day the term “square” meant only “right-angled” — then such a figure was called a perfect square. As oblong figures could also be constructed with other than right angles at the corners, the term “oblong square” — meaning an oblong with square corners — came into use.

Today the word square is used to denominate space enclosed by four lines of equal length and at right angles to each other, anciently the “perfect square” as distinguished from the “oblong square.”

Those “improvers” of the ritual who would substitute parallelogram for oblong square to denominate the form of a lodge would indeed eliminate ancient language in favor of modern English but to be consistent “mote” than should become “may” or “might”!

It is interesting to note that Sir Walter Scott, in Ivanhoe (written in 1819) begins a sentence descriptive of the space in which a tournament was held: “The form of the in-closure was an oblong square.”

Of the geometrical figures in Freemasonry, none is more important, and more generally misunderstood — perhaps non-understood is a better phrase — than the Pythagorean problem, or Forty-Seventh Problem of Euclid.

As stated in geometry, it seems involved and difficult for the non-mathematical reader. In common words it appears much less difficult; the square root of the sum of the squares of the base and perpendicular of any right angled triangle, will be the length of the third dimension of the triangle.

Let us suppose that a line 3 inches long, has at right angles to one end a line 4 inches long. The square of 3 is 9. The square of 4 is 16. The sum of 9 and 16 is 25. The square root of 25 is 5. The line connecting one end of the base and perpendicular of the triangle, then, is 5 inches long. Three, four and five are used here for convenience — the problem is solvable with dimensions of any length in any right angled triangle.

This problem is the tool of the mathematician, the geometer, the engineer, the navigator. By knowing any two dimensions he may obtain the third. By this mathematical law tunnels may be dug from either side of a mountain and meet exactly in the middle. By it distances and heights are measured. By it buildings are constructed, roads engineered, dams built. And by it the Freemason may read that “proof of God” which some demand: by the Forty-Seventh Problem eclipses of sun, moon and stars maybe predicted. Unerring prediction of a future event is unthinkable without a law, a plan. Any law or plan is equally unthinkable without a Law-giver, a Planner.

Hence the importance of the problem to a society predicated upon the Fatherhood of God and the consequent Brotherhood of Man.

While our ritual imputes the discovery of this wonder to Pythagoras (who was born, probably, 582 B.C.) measurements of the pyramids of Egypt, of a much older era, show that the problem must have been much earlier solved.

The triangle, however, has other values in Freemasonry than that developed in the Forty-Seventh Problem.

Triangles are of several varieties of which two only have Masonic significance. These two are the equilateral, in which all three sides are the same length and the right-angled, in which all three sides may be of different lengths, or two sides the same length with one side longer, but in either case with the base and perpendicular forming a right angle.

The triangle was anciently the perfect symbol of Deity. It is the first figure possible with neither beginning or ending, which can be constructed of three equal length measures. Whether by means of sticks from a tree, thongs of skin, lines scratched upon the ground, primitives would soon discover its construction. Even the earliest idea of a Supreme Being was that of a Power that had neither beginning nor ending. Long before early man learned to draw a circle he could make the triangle, which has neither beginning nor ending. Hence its importance in a system that emphasizes the dependence of man on Deity.

In a Masonic lodge the three principal officers form a triangle. Three Lesser Lights are arranged as a triangle, sometimes right-angled, sometimes equilateral, but always a triangle, never in a straight line. The number three appears throughout Symbolic Masonry; three degrees, three circuits in the Master’s Degree, three steps to the East and upon the master’s carpet, three grand pillars, three columns, three, five and seven steps, three principal rounds on a ladder, three Scripture readings — three, three, three, all reminders of the triangle, the earliest symbol of the Supreme Power known to man.

Certain astronomical references in most rituals are concerned with geometrical figures. We are informed that the All-Seeing Eye sees and understands the revolutions of comets. Leave off one letter and the ritual would read evolutions, which would place this reference before knowledge of the fact that comets revolve about the sun. This discovery is credited to Edmond Halley in the year 1682. Our reference, then, to the revolutions of comets shows that it came into the Masonic system within the last three hundred years.

We are told that the north is a place of darkness because King Solomon’s Temple was so far north of the ecliptic that the sun at noon never shone on the north side of it.

The ecliptic is the path through space that the earth travels in a complete revolution (one year) about the sun.

The earth is a ball, slightly flattened at the poles, which revolves about a line passing through the north and south poles once in twenty-four hours.

If this axis were at right angles with the ecliptic, days and nights would be of the same length the year around (with some small variations due to other factors). But the axis of the earth is inclined to the plane of the ecliptic by 23½ degrees. Hence when the axis is inclined towards the sun during the day the days are long; when inclined away from the sun (six months later) the nights are long.

Solomon’s Temple was situated approximately in 31 degrees latitude; hence at high noon in midsummer the sun appeared 7½ degrees south of a point overhead — such as the top of the wall of the Temple. As the sun never shone on the walls of the Temple on the north side, the north became, Masonically “a place of darkness.”

The simplest geometrical figure is a point; that which has position but no dimension. Move a point in a straight path and the result is a line. Move a line in a path parallel to itself and the result is a plane — or, as in most rituals, a superficies. Move the plane in a line at right angles to itself and what is developed is a solid.

The line in Masonry is mentioned in the Fellowcraft Degree, appears also as the plumbline and is symbolized by the twenty-four inch gauge.

The plane is represented by the “plane of the ecliptic” and also by the lodge floor. The solids in our system are especially the rough and perfect ashlars; in ancient rituals the perpend ashlar (a bonding stone) the cornerstones of buildings, and by some students thought to be also named in old rituals as the broached thurnel, a term that has disappeared from our work but that may have been many think must have been — a cubical stone with a pyramidal top.

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NOTE: Readers desiring “more light” on the subjects of any of these thumb-nail sketches are referred to The Short Talk Bulletins entitled: Ancient Square; Corner Stone; 47th Problem; Left to Right; Lesser Lights; Masonic Firmament; Masonic Geometry; Mathematics; Point Within a Circle; Rough and Perfect; Seven Liberal Arts and Sciences; Square, Level and Plumb; Sun, Moon and Stars; “Survey of Nature”; 3-3-7; Three Grand Columns; Three Principal Rounds; Twenty-four Inch Gauge.

The Masonic Service Association of North America