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

Fair and Fowl Correspondences

by Midnight Freemason Guest Contributor

RW Spencer A. Hamann

The thoughtful pupil of Freemasonry cannot help but be pleasantly astounded by the various correspondences, both designed and serendipitous, which permeate our Craft. These corollaries irrevocably link the philosophy of the Craft with the whole of the human experience, if one only deems to pull the thread.

Consider the origin and formation of the first Grand Lodge. There is hardly a Mason who has not read or been told the anecdote: four early 18th Century Lodges in London, England decided that in the interest of unity they ought form a “Grand Lodge” or central regulatory body. On Saint John the Baptist’s day in June of 1717, the very first Grand Master (one Anthony Sayer) was duly installed at the Goose and Gridiron Ale House in Saint Paul’s Church Yard.

For many, the story ends here, relocated to little more than a piece of trivia or a few sentences within a handbook. In truth, there is not a great deal on the written record about this event, save for some accounts both before and after the formation, and a few paragraphs in Anderson’s Constitutions. Besides additional research that can be done upon the climate of early Freemasonry, geography, and some of the men present, there appears to be little more to consider about the event.

Let us momentarily take a step back from this topic, and instead turn our attention to a figure of classical Greek Mythology. Orpheus, son of the musical muse Calliope and Oeagrus, King of Thrace, is particularly noted as a musician of exceptional skill. The music he plays on his lyre (an ancient stringed instrument somewhat akin to a small harp, and a thread which itself leads to fascinating occult musical and mathematical correspondences with Pythagoras) has supernatural abilities to charm humans and nature itself, even overpowering the spell of the Siren’s song.

As the myth goes, Orpheus marries the beautiful Eurydice, who not long after their marriage is bitten by a snake and perishes. Overcome with grief, Orpheus journeys into the underworld where he is met with numerous challenges and obstacles in his attempt to bring back his love. So great is his resolve and skill that he uses his music to persuade Hades to allow him to retrieve Eurydice. However, Hades sets one stipulation: Eurydice may follow Orpheus out of the Underworld, but Orpheus is not to look back or Eurydice will remain in the Underworld forever. Orpheus and Eurydice journey out of the Underworld, but as they are on the brink of escape, Orpheus is overcome with a terrible fear that his wife may not be following him. He turns to see if she is still behind him, and in doing so, both sees her faithfully there and loses her forever.

After Orpheus returns to the world of the living alone, he spends the rest of his days in quiet worship of the Sun (Apollo). There are multiple versions of how Orpheus died, but one widely recounted version expounds that he is murdered, playing his music until the very end. He is then transformed into a swan (a possible corollary to the concept of a “swan song”), and the muses place him and his lyre amongst the stars in the heavens.

Interestingly, in classical Astronomy, the 2nd Century AD astronomer Claudius Ptolemaeus (Ptolemy) included the constellation Cygnus, the swan, amongst his 48 recognized constellations. Ptolemy also lists the constellation Lyra, the lyre, which closely neighbors Cygnus. These constellations both in name and proximity can be viewed as an astronomical representation of the Orpheus myth, and are still visible and recognized today in modern Astronomy.

Further tracing the Liberal Arts, as Western Art Music and specifically opera developed, the Orpheus myth was a favorite of early narratives. It is not difficult to see the appeal: the protagonist is a musician who uses music to supernatural effect during the story, and essentially “dies” and returns from death (the underworld) during the story. In fact the earliest surviving opera, written by Italian composer Jacopo Peri (1561 – 1633) and premiered in 1600, was titled Euridice and is essentially a telling of the Orpheus myth.

Whether fans of the form or not, most of us are already very familiar with a part of an Orpheus-inspired opera: the theme “Galop Infernal” from 19th Century French composer Jacques Offenbach’s Orpheus opera Orphée aux Enfers, which is more lovingly known as a tune and dance craze by the name of the “Can-Can”.

The mythical, astrological, and musical associations with Orpheus have made him a popular figurehead for performing arts troupes over the centuries. Indeed there have been numerous societies and guilds which have taken up the moniker over the centuries, including the Swan & Lyre Musical Society, which has been active for over 500 years in England.

What does any of this have to do with Freemasonry, or the founding of the first Grand Lodge? As many a philosopher and Mason have asserted over the centuries: the answer lies in the pub. The Goose and Gridiron Ale House, a seemingly nonsensical name, is in actuality a parody of Swan and Lyre, the name of a musician’s guild which also met at the public house. Upon examination, the parody name makes perfect sense: a goose is a bird similar in build to a swan, and a lyre with its many strings, when turned sideways does indeed resemble an iron cooking grate (gridiron). While it is unclear if the pub was named Goose and Gridiron from the start, or if it was actually first named Swan and Lyre and became lovingly referred to as the Goose and Gridiron (in a similar way as one may acknowledge a certain global hamburger proprietor as “the golden arches”), by 1717 the pub was widely known as the Goose and Gridiron, and had a prominent sign hung above its facade with the image of a long necked bird and many slatted rectangular shape.

What becomes additionally interesting is the parallel between the myth of Orpheus and the Craft Lodge Masonic legend of Hiram Abiff. While specifics of each story vary, both tell of an extraordinary individual who leaves the realm of the living and rises back up from the realm of the dead to an ultimately higher realm. This transformation and transmutation through death can also be found in the doctrines of many religions and philosophies, including as Christianity and Alchemy.

Whether our early Brethern made a conscious choice based on symbolic correspondences when they chose the Goose and Gridiron Ale House as the birthplace for the first Grand Lodge is a matter of speculation: there has been no known documentation to surface with any evidence to confirm such a claim. However there need not to have been a deliberate and philosophical decision made to bring this together satisfactorily: even coincidences, by definition, are remarkable events, and whether wrought by the hearts and minds of men or drawn on a far greater trestleboard than they can comprehend, the universe is found to be a remarkable place when one only deems to pull the thread.

~SAH

RW Brother Spencer has a vast and varied set of skills and interests, and boundless curiosity. Drawing inspiration from his love of music and engineering at an early age, Spencer began apprenticing to learn the luthier’s art as a high school freshman in 2003. He graduated from the University of Wisconsin-Milwaukee in 2011 with a Bachelor’s of Fine Arts in Musicology, focused on Organology. He now works professionally as a luthier, managing workshops in Illinois and Wisconsin, completing repairs and restorations of fine string instruments, and providing education on instrument history and maintenance. Spencer also enjoys performing music on a variety of instruments, is an avid collector of fine tools, frequently takes commissions for custom work within woodworking, and restores typewriters and bicycles among other antiques. Curatorship and adding value are core to his personal philosophies. Spencer was Raised in 2013, served Libertyville Lodge No. 492 as Worshipful Master from 2017-2018, and currently 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).