The Ecliptic and The Golden Ratio


  In preparing for this presentation, I decided to combine two historical subjects and tie them to the present day.  The references given are contained within the official West Virginia Masonic text book and are “proper to be written”.  Research revealed a great many details about these subjects, which I did not include.  Please feel free to expand these subjects if you wish.  They will certainly provide for a great deal of conversation amongst your Brethren.

Charles E. Kingery PM

Brethren, someone once told me that the secret to a good speech was to have a good beginning, and a good ending; and to have the two as close together as possible.  I’ll try to do that for you today.

Worshipful Master, Wardens and Brethren of                         lodge number________ .

Thank you for inviting me to speak tonight.  There are a several things I want to share with you two of which are, The Beautiful Proportion and The Ecliptic.  As you hear about them they may seem unrelated, but together they will lead us to a third subject which will tie them all together and I hope remind us all of a very important lesson in Freemasonry.

In the first degree lecture, as Entered Apprentices, we learn that the Temple of Solomon was situated North of the Ecliptic, that when the sun or the moon were at their peak, neither could manage to project any light into northern portion of that building.

Now, the Temple of Solomon is in Jerusalem near the equator.  So, it puzzled me that it might be so far North, that the sun at meridian (which is noon) would appear in the South instead of directly overhead.  I wondered why, and what is thing called The Ecliptic?

Researching this I found that The Ecliptic line was first noted by the ancient Greeks.  It is the path the Earth takes as it rotates around the Sun.  It is either north or south of the Equator depending on which hemisphere you are in, because the Earth is not straight up-and-down.  Rather it tilts at about 21.3 degrees.

The Ecliptic is represented on a globe by a diagonal line and on a map by a wavy one.

In the Western Hemisphere it is above the Equator, and in the Eastern Hemisphere it drops below the Equator.

Many people believe that until Galileo, scholars thought the Earth was flat.  Actually, the debate on whether the Earth was round or flat, had been going on for over a thousand years.  In about 300-400bc the ancient Greek scientist and mathematician Eratosthenes (the 3rd chief librarian of the Library at Alexandria) made the observation that in the city of Alexandria, a shadow cast near a well, on one of the solstices at noon, had a given length.  Eratosthenes noted this shadow was particularly short on this day.

Not knowing the significance of the observation, but believing it to be important, Eratosthenes nevertheless made a note of this fact and recorded some measurements.  On the same solstice some years later, after travelling abroad and returning to Alexandria, Eratosthenes stopped at a well in the city of Swett, some 670 miles south of Alexandria, where he saw the sun at noon perfectly framed in the circle of a well.  Eratosthenes looked around and seeing no shadows, immediately recognized the significance of this fact.  Upon returning to Alexandria, Eratosthenes recalled his measurements of the shadows in Alexandria and realized their length represented about 1/500 of the arc of a circle.

From this observation using sacred geometry, Eratosthenes was not only able to determine that the Earth was round, but also to accurately calculate the circumference of the Earth to within 1% of its actual measurement.  He also calculated the distance of the Earth to the Sun.  Eratosthenes not only determined that the Earth revolved around the Sun (rather than the other way around), but he was able to determine that because of the two solstices, the Earth’s orbit was not a circle, but an ellipse.

Now—In everything we do, there are proportions.

In the second degree lecture we learn that architects, geographers & astronomers are enabled to apply geometry to their various purposes.  Today, we all know that science and mathematics go hand-in-hand.  That particular branch of mathematics known as “Geometry” allows us to fix heights & distances, volumes and areas; and is even used in chemistry and physics to determine mass, density and predict the outcome of various reactions.

In the second degree receiving lecture we are taught that geometry can explain “how the planets move in their different orbits and demonstrate their various evolutions.”  We learn that “…we may curiously trace nature to her most concealed recesses”; that we may “…view with delight the proportions which connect this vast machine”.

The beautiful proportion is real.  It is known more commonly as “the Golden Ratio” or 1.6180, and it is all around us.  The golden ratio is expressed in the spirals of a conch shell, the spikes on a pineapple, the branches of a tree and in the rows of spikes in a pine cone.

I’ll bet you didn’t know, that although they may be of different sizes and shapes, each successive row of spikes on a pine cone increases or decreases at a rate of approximately 1.6 times the row before it.

The Greeks were the first to notice this ratio in nature, and began to imitate it in their art and architecture.  This ratio is used most often in the overall dimensions of buildings, the scrollwork on Greek columns and in the number and placement of columns & pilasters for the best structural and aesthetic affect.  Remember, “…it is to the Greeks therefore that we are indebted for what is great, judicious and beautiful in architecture.”

Our ancient friend and brother Pythagoras developed the “Pythagorean Theorem” while studying the golden ratio.  His theorem, A^ + B^ = C^, tells us that the sides of a right triangle are directly related to each other by their lengths.  It also tells us that the degree of each angle and the square area of each side, is directly related to each other.   The ratio between these relationships is… you guessed it, about 1.618.  Many centuries’ later scientists discovered this ratio explains the degree of ellipse and the maximum diameter of each planet’s orbit around the sun.  Incidentally, a perfect ellipse is difficult to draw freehand.  Usually we need a device called a trammel or ellipsograph to help us.  But, an ellipse can also be drawn using a string and two specific instruments together… a square and a compass.   This is the “ tie” we were looking for.

Many renaissance era artists use the golden ratio for the height and width of their paintings, believing this ratio to be the most pleasing to the eye.  The golden ratio was used by Leonardo Da Vinci in his drawing known as “the Vitruvian Man”.  In this drawing we can see that the golden ratio is repeated in the length of one arm across both shoulders to the length of the other arm; or in the length of the lower leg to the length of the upper leg, the length of the head and torso to the length of any leg and so on.  All the different parts of the body, indeed most things in nature are arranged according to the golden ratio just as described in the 2nd degree lectures.  Recently, scientists did a study to see if “beauty” could be mathematically quantified.  They discovered that in fact it can.  They discovered there is a certain proportion for the size and placement of facial features which is most pleasing to the opposite sex, and that perfect proportion is… again you guessed it, is The Golden Ratio, 1.6180.

Even the five pointed star (first used by students of Pythagoras as their symbol), used by the Eastern Star; is a perfect model of the golden ratio.  The ratio between the long side and the short side of the triangle formed by any one of the five points is 1.6180.  The length of two linear segments to the next adjacent segment is 1.6180.

The square area of any three points to the square area of the other two is 1.6180. The area of all 5 points together, to the area of the pentagon in the center is 1.6180; and if you were to draw a diagonal line between any two points bisecting the pentagon, the ratio between the square area of each half would be 1.6180.

Finally, in the third degree lecture there is a portion which describes three steps, emblematic of the three stages of human life; youth, manhood and old age.

This section admonishes us in our youth to attain useful knowledge and in manhood to apply that knowledge in discharging our respective duties so that in old age we can enjoy well spent lives.  Each lecture touches on a different lesson in masonry.  Each is backed by some historical or scientific fact.  When you think about it, isn’t it amazing what the ancients knew?

What I find most interesting, is that while much of the catechism and lectures in masonry are founded in history and science, the most important lessons masonry has to offer are moral ones.

As young masons we should indeed learn all that we can learn, both in the Lodge and in our daily lives.  In manhood as Fellowcrafts we should apply that knowledge so that we may contribute to the greater good of our families or neighbors and ourselves.  Being a Master Mason doesn’t mean we get to sit down and rest either.  As “masters of the Craft” our duties are to help and support our Lodge, in the work we do, in teaching and encouraging young masons, and establishing the foundations of a strong and solid future for our Lodge.

The way we conduct ourselves, the way we help and support each other, the way we behave both in the Lodge and outside the Lodge; are reflected in our lives and are a part of what Masonry has to offer.  It’s often said that the funeral service is one of the very few things we as Freemasons do in the public eye.  But that isn’t quite true is it?  Whether we realize it or not, each of us are representatives of Masonry to the rest of the world.  The way we live our lives in public, our successes, our failures, our weaknesses and our strengths all speak volumes to those who know us to be Freemasons.

Brethren, I’ve heard these words spoken in the Lodge.  They’re not mine, but I have borrowed them, and I’ll leave them with you tonight.  I hope they stay with you and that you consider them often… “I’d rather see a sermon lived than hear one merely spoken.”

So, remember The Ecliptic, The Golden Ratio, and the lessons of Freemasonry.

Thank you Worshipful Master, and thank you my Brethren.