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