 |
| 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 47th 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 47th
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 47th 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.
Tupelo Masonic Lodge No. 318 F&AM
*Home - http://www.tupelomason.org/
*Blog - http://tupelomason.blogspot.com/
*Facebook - http://www.facebook.com/TupeloMason
*Twitter - http://twitter.com/tupelomason
*PHOTOS - http://tupelomason.shutterfly.com/
©2011 All Rights Reserved - TupeloMason®
*Home - http://www.tupelomason.org/
*Blog - http://tupelomason.blogspot.com/
*Facebook - http://www.facebook.com/TupeloMason
*Twitter - http://twitter.com/tupelomason
*PHOTOS - http://tupelomason.shutterfly.com/
©2011 All Rights Reserved - TupeloMason®
