THE DIVINE PROPORTION: TRACING NATURE THROUGH HER VARIOUS WINDINGS: Bro. P.D. Newman

Bro. P.D. Newman
Tupelo Lodge No. 318

Geometry has two great treasures; one is the Theorem of Pythagoras; the other, the [Divine Proportion]. The first we may compare to a measure of gold, the second we may name a precious jewel.[1]

   It is said that, above the threshold of Plato’s Academy at Athens, the words “Let no one ignorant of geometry enter here” were inscribed. As the “first and noblest of sciences,” we know that geometry also serves as an important study for members of the Masonic Fraternity. By it, “we may curiously trace nature through her various windings to her most concealed recesses…and view with delight the proportions which connect this vast machine.” Remarkably, there is one geometrical proportion in particular which should be of special interest to the contemplative Freemason; a proportion which, among other curiosities, actually happens to be known for its “various windings.” We here refer to the so-called divine proportion.[2]

[1] Johannes Kepler

[1] Luca Pacioli, De divina proportione

The divine proportion is related to aesthetics, the Fibonacci sequence, and thus the spiral, the Kepler conjecture, the 47^th problem of Euclid, Penrose tiling, and architecture. Because of its appearance in virtually every kingdom of nature, the divine proportion has been aptly termed by scientists, artists, and mathematicians alike as “the fingerprint of God” [2] as well as the very “face of God.”[3] It appears in the structure of certain minerals, the leafing cycles of plants, the family trees of some insects, and even in the proportions of the human body. For some, this would be but a curious yet simple mathematical coincidence. But, for others, it is veritable evidence of a “Divine Artist;” of a truly “Grand Artificer” who, in executing His design, has left His spiraling mark, not unlike a Mason’s Mark, engraved upon all of the objects and subjects of His creation. In the present article we will examine one symbol from each of the three degrees of Ancient Craft Masonry and demonstrate those symbols’ relation to the divine proportion.

[1] Euclid’s Elements

[1] Mario Livio, The Golden Ratio

[1] James C. Stewart, The Winding Stair: Geometry and the Secrets of Nature

Mention is made in the Entered Apprentice degree of the three ornaments of a Lodge. One of those ornaments is the blazing star[1] which is found in the center of the mosaic pavement. It is said that the blazing star represents “Divine Providence” and, according to one important exposé, “that prudence which ought to appear conspicuous in the conduct of every Mason.”[2] While not specifically noted as being a pentagram, at least ever since the inception of the Order of the Eastern Star in America, it has become commonplace to depict the blazing star as such. In addition to its serving as the jewel for and being depicted on the apron of the Knight of the Sun or Prince Adept degree (27°) of the Ancient and Accepted Scottish Rite, Southern Jurisdiction, the pentagram is commonly associated with ceremonial magic and the goddess Venus, as well as being symbolic of the fifth Sephirah Geburah, the five classical elements (inclusive of spirit) of the ancients, and the Pentagrammaton or name of Jesus interpreted as being Tetragrammaton (YHVH) with the addition of the letter Shin in its midst.[3]

“Amongst the Hebrews, the five point symbol was ascribed to Truth and to the five books of the Pentateuch. In Ancient Greece, it was called the Pentalpha[4]. Pythagoreans considered it an emblem of perfection or the symbol of the human being….[T]he pentagram had a secret significance and power to the Pythagoreans, and was used as a password or symbol of recognition amongst themselves.”[5]

   In the noble science of geometry, the pentagram is intimately involved with the divine proportion. If the pentagon[6] at the center of a pentagram is inlaid with a second pentagram, any given line of the pentagon is in divine proportion to the lines in the smaller pentagram which connect each of the points to one another; any given line of the pentagon is in divine proportion to either one of the two lines composing the apex of any given arm of the smaller pentagram. Similarly, any one of the two lines composing the apex of any given arm of either pentagram is in divine proportion to the base of the triangle which constitutes that same arm of

[1] In the Ancient and Accepted Scottish Rite, Southern Jurisdiction, “[t]he Blazing Star represents the great Central Light, which many nations worshipped in the Sun, its representative.” – Arturo de Hoyos, Scottish Rite Ritual Monitor & Guide

[2] Arturo de Hoyos, Light on Masonry

[3] In the 27° of the Ancient & Accepted Scottish Rite, Southern Jurisdiction, the pentagram is instead related to the Divine Name Elohim and to π.

[4] The Pentalpha is “[t]he triple triangle of Pythagoras, thus called because it shows the form of Alpha – the letter A – in five different positions. The number five formed by the union of the first odd and the first even [numbers] is of peculiar value. It was regarded as a talisman, as a preservative from danger, and, inscribed on the threshold of a door, it kept out evil spirits. It is found almost everywhere in Egypt and Greece, and in the middle ages was greatly esteemed. The early Christians considered that it referred to the five wounds of Christ. In Masonry, it forms the outline of the five-pointed star.” – K.R.H. Mackenzie

[5] http://freemasonry.bcy.ca/grandlodge.html (The Pentagram)

[6] The pentagon is “[a] figure of five sides and five angles, the third figure from the exterior in the camp of the Sublime Princes of the Royal Secret, 32° A. and A. Rite. The pentagon was also used by Cagliostro in his Egyptian Masonry.” – K.R.H. Mackenzie

said pentagram. Additionally, we are told in the lectures that by geometry “we [may] discover how the planets move in their different orbits, and demonstrate their various revolutions.” Remarkably,

“[p]lotting the recurrence of Venus’ westward elongation from the Sun, over six consecutive synodic periods, will create the points of the pentagram…[i.e., the] cyclical positions of Venus will determine the points of a pentagram figure in the morning or evening sky during certain times of the year.”[1]

There are several other examples of how the blazing star (when considered as a pentagram) is related to the divine proportion, but the reader will be left to his own ingenuity to discover those into which space does not here permit us to enter.

  In the Fellowcraft degree we encounter another symbol that is intimately connected to the divine proportion or, in this case, to be more specific, the Fibonacci sequence. I speak here of the winding staircase.[1] The Fibonacci sequence is a series of numbers wherein each successive numeral is the sum of the proceeding two. The first two digits being 1, the Fibonacci sequence progresses as 1, 1, 2, 3, 5, 8, 13, etc., making this series directly applicable to the divine

[1] http://freemasonry.bcy.ca/grandlodge.html (The Pentagram)

[1] “The winding stairs do not begin until the initiate has passed between the pillars of strength and establishment; and there, as a Fellow Craft, he commences the ascent by three, five, and seven [steps], although the Prestonian lectures of the last century give the whole number as thirty-six – divided into one, three, five, seven, nine, and eleven.” – K.R.H. Mackenzie

proportion, being that it is really only an estimated application of the same sequence.[1] Interestingly, Johannes Kepler discovered that the base and perpendicular of the triangle formed by the three boxes which demonstrate the 47^th problem of Euclid, an important symbol in the Master Mason degree, also are in the divine proportion. However, it was Theodorus who discovered that, by applying certain other mathematical theorems to the Kepler triangle which are too complicated to detail here, except to say that Theodorus’ method involves the use of both square roots and the Fibonacci sequence, an image; namely, the spiral of Theodorus[2], may be formed, the same of which actually markedly resembles of all things a winding staircase![3] Perhaps this golden ratio is the precise proportion to which our ritual progenitors allude when we hear of the “beautiful proportions” and “various windings” of nature spoken of in the Fellowcraft degree. It is notable that this same spiraling design can be seen in the shapes of certain seashells which, according to the mysterious alchemist Fulcanelli, were themselves important symbols among early stone masons.[4]

   Lastly, in the Master Mason degree, in addition to the 47^th problem of Euclid, there is another symbol which is connected to the divine proportion and the Fibonacci sequence. I refer here to possibly the most fascinating example of a Masonic symbol as relating to the divine proportion: the beehive.[1] Recall that the Fibonacci sequence progresses as 1, 1, 2, 3, 5, 8, 13, etc. Amazingly, this series of numbers is mirrored in the family tree of male honeybees. For, an

[1] Fibonacci, Liber Abaci

[1] Also known as the Pythagorean spiral

[1] Harry K. Hahn, The Ordered Distribution of Natural Numbers on the Square Root Spiral

[1] Fulcanelli, The Mysteries of the Cathedrals

[1] The beehive is “[a]n emblem of industry; appropriate to the third degree. This virtue is ever held in high esteem in the Craft, for the Old Charges tell us that “all Masons shall work honestly on working days, that they may live creditably on holidays.” The esoteric meaning of the beehive was that of regeneration; and a hive was a type of the ark. “Hence,” says Faber (Orig. of Pag. Idol., vol. ii p. 133), “both the diluvian priestesses and the regenerated souls were called bees; hence bees were feigned to be produced from the carcase of a cow, which also symbolized the ark and, hence, as the great father was esteemed an infernal god, honey was much used both in funeral rites and in the mysteries.” Samson found honey in the carcase of the dead lion. Bees and lilies have many esoteric significations [see Bro. Hargrave Jennings’ The Rosicrucians: Their Rites and Mysteries].” – K.R.H. Mackenzie

unfertilized honeybee larva will always hatch into a male; that is, a drone bee. However, in the event that said larva is fertilized, that larva will then hatch into a female. Therefore, it is only female honeybees which are given the benefit of having two parents, whereas male honeybees are equipped with only one. So, if one traces the family tree of any male honeybee, what he will discover is that said bee will in every case have but one parent, two grandparents, three great grandparents, five great-great grandparents, eight great-great-great grandparents, etc.[1] Thus we see that, in addition to the Masonic beehive being “an emblem of industry,” it also conceals a far more profound mystery which, like the honeycomb hidden away within the hive, while concealed from sight, is discoverable just below the surface.

   The divine proportion is indeed a remarkable mathematical phenomenon. Because of its widespread appearance in nature, it has been considered by various prominent figures to be evidence of a divine Creator and inspiration for artistic creation alike. For, the noble science of geometry is without a doubt “the basis upon which the superstructure of Masonry is erected.” And, of course, it is by geometry that “we [may] discover the power, the wisdom, and the goodness of the Grand Artificer of the Universe.”

REFERENCES

De Hoyos, Arturo Light on Masonry

De Hoyos, Arturo Scottish Rite Ritual Monitor & Guide

Duncan’s Ritual of Freemasonry

Euclid Elements

Fibonacci Liber Abaci

Fulcanelli, The Mysteries of the Cathedrals

Hahn, Harry K. The Ordered Distribution of Natural Numbers on the Square Root Spiral

http://freemasonry.bcy.ca/grandlodge.html (The Pentagram)

http://www.cs4fn.org (Marks for the da Vinci Code: B-)

Livio, Mario The Golden Ratio

Mackenzie, Kenneth R.H. The Royal Masonic Cyclopædia

Pacioli, Luca De divina proportione

Stewart, James C. The Winding Stair: Geometry and the Secrets of Nature

Disclaimer: This paper entitled. "THE DIVINE PROPORTION: TRACING NATURE THROUGH HER VARIOUS WINDINGS"", was submitted to Tupelo Masonic Lodge No. 318 F&AM for publication by the author, P.D. Newman. The printing of this or any other writing does not necessarily reflect the views or opinions of Tupelo Masonic Lodge No. 318 F&AM or the Grand Lodge of Mississippi. Please read our Terms of Use for full details.

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