Cosine Error

by Midnight Freemason Contributor

RWB Spencer Hamann

Imagine you are an operative mason, working with hewn and shaped ashlars. You are asked to check the length of a particular ashlar that is about to be brought up and set into place. Employing your gauge, you take the quick length measurement and communicate it to your counterpart, who confirms it will be an exact fit. The team hoists the ashlar up and into place, but there’s a problem.

The stone is too short.

The length is clearly off, and there are gaps around the sides where it meets with the surrounding stones. Annoyed, your counterpart measures the ashlar’s length and finds it to be different than your measurement; not off by much, but short enough to cause a poor fit. You check your gauge against his, and both instruments are identical.

So what happened? Somehow you took the measurement incorrectly. It is possible that you read the wrong number off of your gauge, but supposing that you read your instrument correctly, the most likely culprit is a trigonometry phenomenon often referred to in modern measuring and layout as “cosine error”.

When taking the measurement of an object of a fixed length, one must naturally measure the correct parameter. In our example of the ashlar’s length, it is the distance straight across a particular rectangular surface of the stone. But what happens if your gauge is not laid exactly parallel to the surface you want to measure? In practice, you are measuring a diagonal across the face of a rectangle. Another way of thinking of this is that by measuring on a slight diagonal, your gauge is creating the hypotenuse of a right triangle.

Our widely taught Masonic symbols include the 47th Problem of Euclid, which is much more than a trifling reminder to appreciate the arts and sciences. While new Master Masons may not have heard the name “47th Problem of Euclid” prior to their Third Degrees, nearly all of us have tested these relationships under a different mathematical name: The Pythagorean Theorem. Within this mathematical proof is an established and fixed relationship between the lengths of the sides of a right triangle, and knowing any two of these dimensions allows us to calculate the third.

Pertaining to our current examination, the Pythagorean Theorem establishes that the hypotenuse of a right triangle is also the triangle’s longest side. Therefore, taking a measurement of any side of a rectangular plane (the surface of an ashlar in this case) with a gauge not held parallel to the side of the rectangle being measured will always result in a longer measurement than the rectangle’s side.

You can better visualize this phenomenon by trying it yourself. Take an ordinary rectangular piece of printer paper, and try to measure the length of one of the longest sides by laying a rule on top of it measuring edge to edge. Now use rule and measure the piece of paper across the opposite corners. You will see that this measurement is longer than the measurement you took of the paper’s length, and the resulting shape is a right triangle with your rule as the hypotenuse. This holds true if you maintain one end of your rule in contact with the corner, and use it as a pivot to swing across the surface of the paper. The distance the rule measures will decrease as it gets closer to parallel with the side of the piece of paper, but it still measures longer than the piece of paper itself.

So where does “cosine error” enter into this? If we following the trigonometry even further, we also know that there is a mathematical relationship between the lengths of the sides of a right triangle and the angles that they form. When we measure the hypotenuse of the formed right triangle, the measurement we are taking is essentially the length of the rectangle’s side we wanted to measure with the cosine of the angle it forms with that side included. Inadvertently, we are introducing additional trigonometric complexity into what should have been a straightforward measurement had we only used our gauge correctly in the first place.

Cosine error is a possibility anytime one is taking a dimensional measurement parallel to a plane, and can be of serious consequence in high-precision work where close tolerances matter. However, the name can be a little misleading: the error is really that of the craftsman taking the measurement, not the accidentally introduced trigonometry. Measuring tools don’t know if they are being used correctly, and will display information regardless of how they are being employed. It is the mastery of a tool that allows it to perform the function it was designed for and become a help rather than a hindrance.

This same idea applies to our speculative working tools. It is not enough simply to be invested with a 24 inch gauge: a craftsman must understand how to use it and furthermore actually put it to use. It is only through work that these principles and lessons represented allegorically by our working tools come to life and give us any real benefit. In the case of our 24 inch gauge, if we are not carefully and purposely planning out and dividing our time, we quickly find that we are either coming up short or over extending ourselves. Either of these results can prove disastrous not only to our own mental and personal well-being, but for those we love and our obligations owed.

Mastery comes only through mindful practice, contemplation of refinement, and a desire to improve. Without these essential characteristics, even though we may be attempting to put our working tools to task we are inadvertently introducing error into our work. If we cannot identify and correct the errors built into our own edifices, we stand no chance of meaningfully sharing the light we seek to create in the world around us.

RWB Spencer Hamann is a luthier and musicologist working in northern Illinois. He is an avid woodworker and artificer and enjoys antique restorations and custom commissions. Curatorship and adding value are core to his personal philosophies. Spencer was Raised in 2013 and served Libertyville Lodge No. 492 as Worshipful Master from 2017-2018. He is the Senior Warden of Spes Novum Lodge No. 1183, and serves the Grand Lodge of Illinois as their Grand Representative to Wisconsin, District Education officer for the 1st NE District, and is a Certified Lodge Instructor (CLI). He can be contacted at spencer@sahamann.com

The Circumpunct

by Midnight Freemason Contributor

WB Darin A. Lahners

In the First Degree ritual, Freemasons are taught about the point within a circle or the circumpunct.  The uncyphered Illinois ritual states: "Lodges in ancient times were dedicated to King Solomon, he being our first Most Excellent Grand Master; in modern times to St. John the Baptist and St. John the Evangelist - two eminent Christian patrons of Freemasonry; and since their time there has been represented, in every regular and well furnished lodge, a certain point within a circle, embordered by two perpendicular parallel lines, representing St. John the Baptist and St. John the Evangelist.  On top of the circle rests the book of Holy Scriptures; the point represents an individual brother, the circle the boundary line of his duty. In going round this circle we necessarily touch on the two parallel lines, as well as on the book of Holy Scriptures; and while a Mason keeps himself circumscribed within their precepts, it is impossible that he should materially err."  What is not told to you, is that the point within the circle has further allusions.

The ancient alchemical symbol for gold was often represented by the point within the circle.  In Astrology, the Sun is also represented by the point within the circle. Pythagoras, who we are taught was the inventor of the 47th problem of Euclid, was "In his travels through Asia, Africa, and Europe, was initiated into several orders of Priesthood, and is said to have been raised to the sublime degree of Master Mason.  This wise philosopher enriched his mind abundantly in a general knowledge of things, more especially in geometry, or Masonry. On this subject he drew out many problems and theorems; and among the most distinguished he erected this, when, in the joy of his heart, he exclaimed, Eureka! signifying in the Grecian language, I have found it; and upon the discovery of which he is said to have sacrificed a hecatomb."  What we are not told is that Pythagoras used this symbol to represent the Monad.  

The Monad represented to Pythagoras the point of the beginning of creation, the Supreme Being, divinity or the totality of all things.  It symbolizes an idea of cosmic consciousness, a consciousness that experiences itself and learns from itself by observation and experience of the micro and macro, as well as the macro and micro.  The idea of the Monad parallels that which is above Kether, which is the topmost sephirot of the Tree of Life in Kabbalah.

Prior to Creation, there was only the infinite Or Ein Sof filling all existence. When it arose in G-d's Will to create worlds and emanate the emanated ... He contracted (in Hebrew "tzimtzum") Himself in the point at the center, in the very center of His light. He restricted that light, distancing it to the sides surrounding the central point, so that there remained a void, a hollow empty space, away from the central point ... After this tzimtzum ... He drew down from the Or Ein Sof a single straight line [of light] from His light surrounding [the void] from above to below [into the void], and it chained down descending into that void. ... In the space of that void He emanated, created, formed and made all the worlds.

-Etz Chaim, Arizal, Heichal A"K, anaf 2

The Zohar states that Kether is the most hidden of all things. According to Arthur Green in his work, Guide to the Zohar, it represents the primal stirrings of intent in the Ein Soph (Infinity) or the arousal of desire to come forth into the various states of being.  Kether is associated with the name that God gives to Moses, Ehyeh Asher Ehyeh which is commonly translated as "I am that I am" in Exodus 3:14 in the King James Version of the Old Testament. "And God said unto Moses, I AM THAT I AM: and he said, Thus shalt thou say unto the children of Israel, I AM hath sent me unto you."  However, Ehyeh Asher Ehyeh can also be translated as: "I am who I am," "I will become what I choose to become," "I am what I am," "I will be what I will be," "I create what(ever) I create," or "I am the Existing One."  As Kether is the first emanation, the Zohar makes it clear that "the supernal crown (keter elyon) is the crown of kingdom (keter malchut)." Meaning that the first, highest emanation of the Divine- Kether is linked to the last Malchut, a concept which was summarized by Hermes Trismegistus:  "As above, so below; as below, so above."

It is probably not an accident that the name of God is revealed in Exodus 3:14.  Pi is defined as the ratio of a circle's circumference to its diameter.  It is most commonly rounded to 3.14, although its decimal representation never ends nor does it settle into a permanently repeating pattern.  As God is infinite and eternal, so is the numeric representation of Pi.  

To close, I will use a story that I ran across while researching this article (that had no attributable source) to illustrate my last point.  The story goes: 

I had a professor once who had asked the class to draw the best representation of God. There were some students who drew Christ, others an old man in a chair, and then there were some that drew the Hindu gods. In the end, all he did was draw a circle on the board with a dot in the center.“This is God, the circumpunct is perhaps the most perfect symbol to represent God. The circle,” His finger traced the circle on the board. “represents the all-encompassing power and infinite limitlessness of God. While the dot,” He pointed to the center of the circle “Represents our place within God, we are part of God. The circumpunct represents the perfect union of the divine and the created. The dot can represent anything within the omnipotent divinity of God.” He walked around the room in a circle then stood in the center. “Or it can also represent the self. The circle can mean the body, or the conscious self, while the dot can represent our subconscious or super-conscious.”

~DAL

WB Darin A. Lahners is our co-managing Editor.  He is a Past Master of St. Joseph Lodge No.970 in St. Joseph. He is also a plural member of Homer Lodge No. 199 (IL), where he is also a Past Master. He’s a member of the Scottish Rite Valley of Danville, a charter member of Illinois Royal Arch Chapter, Admiration Chapter No. 282 and is the current Secretary of the Illini High Twelve Club No. 768 in Champaign – Urbana (IL). You can reach him by email at darin.lahners@gmail.com. 

Is the Third Degree a teletē?

by Midnight Freemason Contributor

WB Darin A. Lahners   

In Orphic cosmology, the universe was conceived of in a cosmic egg. The idea is that the cosmos was initially a self-contained embryo, which at a certain point hatched, the upper half of the egg formed the Heavens, the lower half formed the Earth. After that ensuing Chaos, the three realms consisting of Heaven, Earth, and Sea, were bound by Aether. This substance which was described as the fifth element by Aristotle, held the three realms together, creating the universe. The Orphics believed in an omnipotent creator, a demiurge, named Phanes. Phanes was the god of all gods until he was devoured by Zeus.  This creation story parallels those of other ancient civilizations. 

Orphism was named after Orpheus, a mythical hero, who tried to retrieve his wife from the Underworld under the condition that he not look at her. He failed and was killed by mænads, who were followers of Dionysus. The religious foundations derive from the myth of Dionysus. Born to Zeus and Persephone, Dionysus was dismembered and eaten by the Titans. An angered Zeus struck the Titans with his thunderbolt, disintegrating them, and reviving from their ashes a reincarnated Dionysus, along with mankind.  

Man had a dual nature, a soul from Dionysus which contained the pure divine spark, and body from the Titans which was impure. For this reason, the Orphics thought the body (soma) was a tomb (sema). To the Orphics, the body reminded man of his corporeal nature, as opposed to his spiritual one. In his dialogue, , Gorgias, Plato states: “I have heard a philosopher [Pythagoras] say that at this moment we are actually dead and that the body is our tomb…” In order to achieve salvation from the tomb of our material existence, one had to be initiated into the Dionysian mysteries and undergo teletē. Teletē is a ritual purification which consisted of reliving the suffering and death of the god. Orphics believed that they would, after death, spend eternity alongside Orpheus. The uninitiated (amúētos), they believed, would be reincarnated indefinitely.  This idea is similar to the Buddhist idea of Samsara, the wheel of rebirth.  If, when we die, we have not balanced our Karma, we are condemned to live another life.  This process would continue ad infinitum until we do so.

By this point, you might be wondering, What does this have to do with Freemasonry?  Pythagoreanism, the philosophical brotherhood started by Pythagoras, took many of the orphic doctrines and incorporated them into his teachings. In the third Degree lecture, we are taught that Pythagoras was the inventor of the forty-seventh problem of Euclid.  We are told that in his travels through Asia, Africa, and Europe he was initiated into several orders of Priesthood, and is said to have been raised to the sublime degree of Master Mason. Historians accredit Pythagoras with being the first to call himself a philosopher; as he considered it a way of life. He believed that philosophy was a life that was lived in discussion and in experimentation. He is said to have “intellectualized” Orphism by applying scientific thinking and reasoning to its beliefs. In doing so, he made it a viable way of life, rather than a mystery. In Gorgias, Plato states: “[T]hey say that the soul of man is immortal, and at one time has an end, which is termed dying, and at another time is born again, but never destroyed. And the moral is, that man ought to live always in perfect holiness.”

Plato describes the ethical system of Orphism, explaining the idea of purification, which is required if one wants to reach the afterlife. This system heavily influences the idea of the immortality of the soul advocated by Pythagoras. It is believed that this is an idea that Socrates taught him and that would play a crucial role in his philosophy. Pythagoras believes that the soul exists eternally and can never be destroyed; if it is impure at the end of its body’s life, it would be reincarnated into the body of a new person, this cycle continuing until at death, the soul was pure, at which point it would go onto be one with divinity. For Pythagoreans, a virtuous life was one that would lead to this purification.  This life consisted of dutiful moral responsibility and severe self-discipline and abstention from all forms of indulgence. Although our bodies were a tomb, the Pythagorean believed that the soul was holy and needed to be pure if it wanted to return to divinity. The life of the Pythagorean was dedicated to caring for and tending to the soul.  They were cautious not to commit any vices. In the afterlife, the soul would be judged by its scars, not the physical ones, but the spiritual ones. These spiritual scars are symbolic of the vices of which the body was guilty. Because of this belief, the soul was of absolute importance and it took precedence over the well-being of the body.

 As I stated, One of the key beliefs of the Pythagoreans was the transmigration of the soul borrowed from the Orphics and expanded upon. In fact, one story about Pythagoras was recanted by Xenophanes. He recalled: “Once they said that he [Pythagoras] was passing by when a puppy was being whipped, and he took pity and said: Stop, do not beat it; for it is the soul of a friend that I recognized when I heard it giving tongue.” In this story, Pythagoras remembered the voice of a friend of his and reasoned that his soul must have been reincarnated as a dog. Pythagoras was famously a vegetarian and anyone who joined the Pythagoreans would become a vegetarian.  This was not for ethical reasons, it was based upon the basis that animals could be the host of either a friend’s or an ancestor’s soul. Beans were also to be refrained from, for Pythagoras said they were the seeds from which humans were birthed. To the Pythagoreans, to eat a bean was to eat a fellow human. Ironically, according to legend, Pythagoras died because he was chased to a bean field by an angry mob, and in not wanting to trample the beans, decided to surrender himself to the mob instead. 

The eating of meat or beans was called Adikia, and it was one of the greatest vices. Plato stated in his work, Laws, that “[M]en are said to have lived a sort of Orphic life, having the use of all lifeless things, but abstaining from all living things.” Orphism was practically synonymous with vegetarianism as a result. The Pythagoreans believed all life was interconnected like a web which was connected to the Divine, of which all living things were a part. A story in Pythagorean teaching tells of a man named Æthalides. He was bestowed by Hermes the gift of being able to remember his past lives. Upon passing, he was reincarnated as Euphorbus, who was slain by Menelaus in the Trojan War. His soul then went to inhabit Hermotimus, who went to a temple and allegedly pointed out the shield used by Menelaus.  In doing so, proving he was Euphorbus in his previous life. Then, Hermotimus died and became Pyrrhus; and finally, the soul went on to inhabit Pythagoras. Because of this, Pythagoras taught his followers to every night to go through their previous day in their memory. In recalling as much detail as possible, as a way of strengthening their memory, they would be able to eventually remember as far back as their own previous lives.

Prior to the second section of the third degree in Illinois, we read code 365A, which strictly forbids "any levity, horseplay or roughness and insists that there be no such actions and no audible laughter or other noise in the Lodge room which might distract the attention of the candidate. Failure to comply with this Code, and any action by any officer or member in violation of or inconsistent with the language of this order shall constitute grounds for disciplinary action."  But what I find really interesting is what is written right before that statement in the code.  I have bolded the sentence that stands out.  "The second section of the Third Degree constitutes a most solemn and impressive portion of our ritualistic work. In it we are taught the ultimate lessons of Masonic philosophy--victory over death and the immortality of the soul. Nothing must be allowed to impair the deep impression which should be made upon the mind of the candidate."  This makes me wonder if Pythagoras and his teachings still resonate within the second section of the third degree?

In the first degree lecture, we are taught the ornaments of the lodge, one of which is the mosaic pavement.  A mosaic pavement consists of several stones of mixed colors joined together in a pattern to imitate a painting.  In Freemasonry, the pavement is depicted as alternating black and white tiles like that of a chessboard.   We are told that it is emblematical of human life, checkered with good and evil.  Does this dualistic philosophy not remind you of the Orphic/Pythagorean belief?  The belief that man has both the divine spark which is pure (good), and the body which is impure (evil).  

If this is the case we can think of the third degree as a teletē.  If the ultimate lessons of Masonic philosophy are - victory over death and the immortality of the soul, then does it not stand to reason that what happens during the second section of the third degree is a ritual purification which consists of reliving the suffering and death of our Grandmaster Hiram Abiff.  I want to be very clear, while the Orphics did this for a god, Dionysus; Freemasons do not worship or deify Hiram Abiff.  That being said, in undergoing this ritual and in following the teachings of Freemasonry, are we not purifying ourselves?   Is the idea of becoming a perfect ashlar not an idea of becoming pure?   By undergoing this ritual in the second section of the third degree, we are taking a necessary final step in the purification of ourselves.  

I believe that while it is not implied nor can it be proven, that the ancient mysteries have heavily influenced our rituals and practices.  Pythagoras is specifically mentioned as our ancient friend and brother in our third-degree lecture for his invention of the forty-seventh problem of Euclid.  Does it not stand to reason that some of his other teachings would have influenced our ritual?  There are other connections to the Egyptian mysteries and others that I will not go into today, but if you look hard enough you will be able to make the connections yourself.   Just don't think too much about whether or not to eat those green beans at the next dinner you are able to have with your brethren after this pandemic is over. 

~DAL       

WB Darin A. Lahners is our co-managing Editor.  He is a Past Master of and Worshipful Master of St. Joseph Lodge No.970 in St. Joseph. He is also a plural member of Homer Lodge No. 199 (IL), where he is also a Past Master. He’s a member of the Scottish Rite Valley of Danville, a charter member of Illinois Royal Arch Chapter, Admiration Chapter No. 282 and is the current Secretary of the Illini High Twelve Club No. 768 in Champaign – Urbana (IL). You can reach him by email at darin.lahners@gmail.com