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

William Schaw and the Schaw Statutes

by Midnight Freemason Contributor

Steven L. Harrison, 33°, FMLR

William Schaw, born sometime around 1550, is best known as the Great Master of Works to James VI of Scotland. As such he was responsible for overseeing all royal castles and palaces. Having the complete trust of the King, he also served him in several other capacities, including accompanying him to Denmark to return with the new queen, Anna of Denmark.

Within Freemasonry Schaw is best known for setting forth the first and second set of Schaw Statutes to be observed by all the Master Masons within the realm. He issued the first set in 1598. Written as a part of his responsibility as Master of works, he directed both statutes primarily to operative Masons; however, these are among the first documents alluding to the esoteric and speculative aspects of the craft.

The first set of statutes specifies 22 rules and regulations designed to govern the work and behavior of Master Masons and their apprentices. Many of the rules set forth a penalty for disobedience. These regulations call on all Masons to "observe and keep all the good ordinances established before... to be true to one another and live charitably together... and be honest, faithful, and diligent in their calling." In it he admonishes masons never to undertake work they can't perform nor take away another master's work. He limits the number of apprentices in a lifetime to three and prohibits the selling of apprentices to other masters. Schaw also outlines rules governing the resolution of grievances and stipulates penalties collected shall go to charity.

The Second Schaw Statute, written in 1599, establishes in order, Edinburgh, Kilwinning and Stirling as the principal lodges in Scotland. It establishes the election of the wardens, deacons and secretaries and some of their duties. It reaffirms the use of fines for charity and dictates exclusion of all who fail to live up to the statutes. It requires every fellow of the craft and apprentice to demonstrate their skills annually and forbids association with cowens.

The inscription on his tomb bears the most reliable source of his biographical information and reads:
"This humble structure of stones covers a man of excellent skill, notable probity, singular integrity of life, adorned with the greatest of virtues – William Schaw, Master of the King's Works, President of the Sacred Ceremonies, and the Queen's Chamberlain. He died 18th April, 1602.

Among the living he dwelt fifty-two years; he had traveled in France and many other Kingdoms, for the improvement of his mind; he wanted no liberal training; was most skillful in architecture; was early recommended to great persons for the singular gifts of his mind; and was not only unwearied and tireless in labours and business, but constantly active and vigorous, and was most dear to every good man who knew him. He was born to do good offices, and thereby to gain the hearts of men; now he lives eternally with God."

His tomb also bears what may be the earliest mason's mark. It is a complex sculpture of all the letters of his name, an S-C-H-A-W, inscribed over a square and compasses. His epitaph concludes, "Queen Anne ordered this monument to be erected to the memory of this most excellent and most upright man, lest his virtues, worthy of eternal commendation, should pass away with the death of his body."

~SLH

Bro. Steve Harrison, 33°, is Past Master of Liberty Lodge #31, Liberty, Missouri. He is also a Fellow and Past Master of the Missouri Lodge of Research. Among his other Masonic memberships is the St. Joseph Missouri Valley of the Scottish Rite, Liberty York Rite bodies, and Moila Shrine. He is also a member and Past Dean of the DeMolay Legion of Honor. Brother Harrison is a regular contributor to the Midnight Freemasons blog as well as several other Masonic publications. Brother Steve was Editor of the Missouri Freemason magazine for a decade and is a regular contributor to the Whence Came You podcast. Born in Indiana, he has a Master's Degree from Indiana University and is retired from a 35-year career in information technology. Steve and his wife Carolyn reside in northwest Missouri. He is the author of dozens of magazine articles and three books: Freemasonry Crosses the Mississippi, Freemasons — Tales From the Craft and Freemasons at Oak Island.

Was Jesus an Operative Mason?

by Midnight Freemason Contributor 

Steven L. Harrison, 33°, FMLR

Coming to Nazareth during his ministry, Jesus preached just as he had been doing throughout the countryside. In other places he had drawn enthusiastic crowds. In this his hometown, however, people in the crowd became derisive. They recognized him to be one of their own, a "common" tradesman, and therefore not someone who should be taken seriously as a teacher or prophet. "Is not this," they asked, "the carpenter?"

This passage is where we learn Jesus, like his earthly father Joseph, was a carpenter, according to contemporary Bible translations. Both Mark 6:3 and Matthew 13:55 give an account of the incident using the Greek word "tektōn" to refer to Jesus' profession.

"Is not this the carpenter ["ho tektōn"], the son of Mary, the brother of James, and Joses, and of Juda, and Simon? And are not his sisters here with us?" ~Mark 6:3 (KJV)

"Is not this the carpenter's son ["ho tou tektōnos huios"]? Is not his mother called Mary? and his brethren, James, and Joses, and Simon, and Judas?" ~Matthew 13:55 (KJV)

The word "tekton," however, does not directly translate as "carpenter." It more accurately means "builder" or "craftsman," a designation which, in fact, leaves Jesus' true profession in doubt.

A carpenter would fit in the category of builder or craftsman, and that may well have been what Jesus did. He would not have built homes in an area where trees were scarce, but would have made furniture, doors and tools, such as plows.

However, the most dominant profession around Nazareth, where Jesus grew up and would have practiced his trade, was stone masonry. The area was rich in stone with several quarries, including one in the heart of Nazareth. Virtually all buildings were made of stone, and the demand for stone masons would have been high.

During the time Jesus would have worked as a craftsman, the Romans expanded the small town of Sepphoris into a city for Jewish aristocrats who supported Rome. True, the venture would have required carpenters, but the greatest demand would have been for masons. Sepphoris (today Zippori) was less than four miles from Jesus' home and, regardless of his craft, it is likely he worked on the project.

We’ll never know for sure since the broad definition of "tekton" could refer to a number of professions. But taken in context and in light of the more likely profession of the tradesmen in Nazareth in that era, it could be that Jesus was not a carpenter, but an operative mason.

~SLH

Investigations

by Midnight Freemason Contributor

RWB Michael H. Shirley

Jurisdictions may vary in the way they do this, but when a man petitions a lodge for degrees, that lodge is required to investigate him, in order to determine whether he is fit to be made a Mason. Some lodges conduct their investigations with due diligence. Others, to put it kindly, do not. I sometimes hear my Brethren lamenting this state of affairs, and harking back to the days when “they used to do it better.”

In 1910, Brother Albert W. Ashley, Most Worshipful Grand Master of Ancient Free and Accepted Masons of the State of Illinois, was moved to include a section on investigations in his report to the Grand Lodge. Entitled “Greater Care Needed,” it read as follows:

Information has come to me from various sources that in many lodges there is great carelessness in investigating the character of petitioners for the degrees. In the inordinate desire for a large membership unfit men are received. In localities where there is a shifting population this is particularly true.

In the oil region complaint is made that candidates have been accepted and the degrees conferred upon those who were mere transients and whose residence is in another state. Others were accepted and the degrees conferred upon who [sic] were morally unfit to he made Masons, and whose occupations violated the law of the state. Such violations of law must inevitably lead to disaster and discipline.

Attention is here called to this evil that Masters and others may take due notice and govern themselves accordingly.

Lax investigations, it seems, are not a new problem, which isn’t surprising. If history teaches us anything, it’s that human beings weren’t nobler in the past, or smarter, or harder working. They were subject to the same failures of character as we are. They just wore different clothes.

~MHS

R.W.B. Michael H. Shirley serves the Grand Lodge of Illinois, A.F. & A.M, as Leadership Development Chairman and Assistant Area Deputy Grand Master of the Eastern Area. A Certified Lodge Instructor, he is a Past Master and Life Member of Tuscola Lodge No. 332 and a plural member of Island City Lodge No. 330, F & AM, in Minocqua, Wisconsin. He currently serves the Valley of Danville, AASR, as Most Wise Master of the George E. Burow Chapter of Rose Croix; he is also a member of the Illinois Lodge of Research, the York Rite, Eastern Illinois Council No. 356 Allied Masonic Degrees, Eastern Star, Illini High Twelve, and the Tall Cedars of Lebanon. The author of several articles on British history, he teaches at Eastern Illinois University.You can contact him at: m.h.shirley@gmail.com